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Working session: Additive combinatorics, entropy, and fractal geometry. Abstracts from the working session held October 8–13, 2017. (Arbeitsgemeinschaft: Additive combinatorics, entropy, and fractal geometry.) (English) Zbl 1409.00074
Summary: The aim of the workshop was to survey recent developments in fractal geometry, specifically those related to projections and slices of planar self-similar sets, and dimension and absolute continuity of self-similar measures on the line, in particular Bernoulli convolutions. The methods combine ergodic theory, additive combinatorics, and algebraic number theory. Talks were high-level descriptions of the results, aimed at a mixed audience with minimal background in real analysis, ergodic theory and dimension theory.
00B05 Collections of abstracts of lectures
00B25 Proceedings of conferences of miscellaneous specific interest
28A80 Fractals
28A75 Length, area, volume, other geometric measure theory
37A25 Ergodicity, mixing, rates of mixing
28-06 Proceedings, conferences, collections, etc. pertaining to measure and integration
11-06 Proceedings, conferences, collections, etc. pertaining to number theory
Full Text: DOI
[1] K. Falconer, Fractal geometry: mathematical foundations and applications, John Wiley & Sons, 2004 · Zbl 0689.28003
[2] M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. 180, 773-822, (2014). · Zbl 1337.28015
[3] M. Hochman, P. Shmerkin, Local entropy averages and projections of fractal measures, Annals of Mathematics, 175 1001-1059. · Zbl 1251.28008
[4] P. Mattila,Geometry of sets and measures in Euclidean spaces, Cambridge University Press, vol. 44, (1995). · Zbl 0819.28004
[5] MJ. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proceedings of the London Mathematical Societ 3 (1954), 257–302. · Zbl 0056.05504
[6] P. Shmerkin, On Furstenberg’s intersection conjecture, self-similar measures, and the Lq norms of convolutions, Preprint 2016.
[7] J. Bourgain, On the Erd˝os-Volkmann and Katz-Tao ring conjectures, Geom. Funct. Anal. 13(2003), 334–365.
[8] J. Bourgain, The discretized sum-product and projection theorems, J. Anal. Math. 112 (2010), 193–1236. · Zbl 1234.11012
[9] W. T. Gowers, A new proof of Szemer´edi’s theorem for arithmetic progressions of length four., Geom. Funct. Anal. 8 (1998), 529–551. · Zbl 0907.11005
[10] A. Iosevich, O. Roche-Newton, M Rudnev, On discrete values of bilinear forms, preprint. · Zbl 1406.52034
[11] N. Katz, T. Tao, Some connections between the Falconer and Furstenburg conjectures, New York J. Math. 7 (2001), 148–187.
[12] S. V. Konyagin, I. D. Shkredov, New results on sum-products in R, Tr. Mat. Inst. Steklova 294(2016), 87–98.
[13] J. Solymosi, Bounding multiplicative energy by the sumset, Adv. Math. 222 (2009), 402– 4088. Introduction to Entropy Agamemnon Zafeiropoulos We define the notion of entropy of a probability measure with respect to a partition, as well as the entropy dimension of a probability measure. We investigate connections between entropy dimension and other notions of dimension. 1. Definitions and Basic Properties Let p = (pi)ibe a probability vector. The entropy of p is defined by X H(p) =−pilog pi. The entropy of a probability measure µ with respect to the partition α of the underlying space X is defined to be X H(µ, α) =−µ(A) log µ(A) . A∈α The conditional entropy of µ with respect to a partition α given the partition β is defined by X H(µ, α|β) =µ(B)H(µB, α) , B∈β Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2855 where µBdenotes the normalised restriction of µ on the set B. Furthermore, given two partitions α, β their join α∨ β is defined to be the coarsest common refinement, i.e. α∨ β = A ∩ B : A ∈ α, B ∈ β . The entropy as defined above satisfies the following properties: • 0 ≤ H(µ, α) ≤ log |α|, with H(µ, α) = 0 iff α is a trivial partition and H(µ, α) = log|α| iff µ(A) = 1/|α| for all A ∈ α. • H(µ, α ∨ β) = H(µ, α) + H(µ, β|α). • H(µ, α ∨ β) ≤ H(µ, α) + H(µ, β). • If µ, ν are probability measures and 0 < λ < 1, then H(λµ + (1− λ)ν, α) ≥ λH(µ, α) + (1 − λ)H(ν, α). • If p = (pi)ki=1is a probability vector and µ1, . . . , µkare probability measures, then !Xk Xk Hpiµi, α≤piH(µi, α) + H(p). i=1i=1 2. Entropy Dimension From now on the positive integer d≥ 1 is considered fixed. We define the n-th level dyadic partition of Rdto be k Dn=1,k1+ 1× . . . ×kd,kd+ 1: k 2n2n2n2n1, . . . , kd∈ Z. The n-th scale entropy of a probability measure µ is defined to be 1 Hn(µ) =H(µ,Dn). n Finally, we define the entropy dimension of µ by dimeµ = limHn(µ), n→∞ provided the limit exists. Whenever dimeµ exists, it is a number in [0, d]. The following proposition shows the relation between the entropy dimension of a measure and the box dimension of its support set. Proposition 1. Let µ be a probability measure. Then dimeµ≤ dimBsupp(µ), provided both dimensions exist. The following theorem shows the connection between pointwise dimension and entropy dimension of a probability measure. 2856Oberwolfach Report 47/2017 Theorem 1. Let µ be a probability measure which is compactly supported in Rd. If µ is exact dimensional with dimension α almost everywhere, then dimeµ = α. More generally, if the pointwise dimension of µ at the point x∈ supp(µ) is α(x), thenZ dimeµ =α(x)dµ(x). 3. Entropy Dimension of Self-Similar Measures Apart from exact-dimensional measures, entropy dimension also exists for certain self-similar measures. Theorem 2. Let Φ =φii∈Ibe an Iterated Function System of similarities in Rdand µ =Pi∈Ipiφi∗µ be a self-similar measure. The entropy dimension dimeµ exists. References
[14] Ai-Hua Fan, Ka-Sing Lau, Hui Rao: Relationships between different dimensions of a measure Monatsch. Math, 135(3): 191-201, 2002 · Zbl 0996.28001
[15] Y. Peres, B. Solomyak: Existence of LqDimensions and Entropy Dimension for Selfconformal Measures. Indiana University Mathematics Journal, 49(4), 1603-1621 (2000). Retrieved from http://www.jstor.org/stable/24901114 · Zbl 0978.28004
[16] Peter Walters: An Introduction to Ergodic Theory volume 79 of graduate Texts in Mathematics, Springer-Verlag, New York, 1982 CP Processes Daniel Glasscock A CP process is, roughly speaking, a measure preserving dynamical system on a space of probability measures under zoom–and–scale dynamics. Harry Furstenberg [1] introduced CP processes in 1970 as a tool in the study of the relationship between 2x and 3x (mod 1) dynamics, and they were recently employed to resolve some of Furstenberg’s original conjectures [4, 6] concerning the dimension of projections and slices of product sets invariant under those dynamics. In this talk, we define CP processes, give some basic examples and properties, and outline in broad strokes the way in which they are used. Dynamics comes to bear on problems in fractal geometry via the repeated action of zooming in on part of a probability measure. To zoom in on µ∈ P([0, 1]) on an interval I⊆ [0, 1] for which µ(I) > 0, we restrict µ to I, push µ forward through the unique homothety which sends I to [0, 1], and renormalize so that this pushforward becomes a probability measure. The goal is to gain insight into the fine-scale structure of µ by repeatedly zooming in and understanding, for example, the trajectory of µ throughP([0, 1]). We can realize this goal by constructing a CP process related to µ and transferring nice properties of that process back to µ. CP processes may be described dynamically as measure preserving systems or probabilistically as random processes. We shall use dynamical language, following Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2857
[17] ; for an introduction to CP processes in the language of random walks and Markov chains, see [3, Section 6] . 1. CP-processes on trees We will define CP processes on symbolic trees. The symbolic setting is helpful because the zoom–and–scale map on measures on a totally disconnected space is continuous. Passing results back and forth between the Euclidean setting and the symbolic setting has its own complications, but we will not address those here. Fix b≥ 2 and d ≥ 1, and let Λ = 0, . . . , b − 1d. Denote by σ : ΛN→ ΛNthe left shift (σw)n= wn+1. For v∈ Λn, let [v] =w ∈ ΛN| w1· · · wn= v. For µ∈ P(ΛN) and v∈ Λnfor which µ[v] > 0, define the measure µv∈ P(ΛN) by zooming in on µ on [v] and scaling: σn∗(µ[v]) µv=, that is, for all cylinder sets [u]⊆ ΛN, µv[u] =µ[vu]. µ[v]µ[v] P∞ The geometric coding map γ : ΛN→ [0, 1]ddefined by w7→n=1wn/bnconnects the symbolic and Euclidean settings; the measure µvcorresponds to the Euclidean measure gotten by zooming in on γ∗µ on the b-adic cube γ[v] and scaling. CP processes will be defined to be measure preserving dynamical systems on a subset of X =P(ΛN)× ΛN, the space of pairs of a measure on which to zoom in and a point indicating where to zoom. EndowingP(ΛN) with the weak-∗ topology, the set X is a compact and metrizable topological space. On the subset X =(µ, w)∈ X µ[w1· · · wn] > 0 for all n∈ N=(µ, w)∈ X | w ∈ suppµ, we define the zoom–and–scale map T : X→ X by T (µ, w) = µw1, σw. A (base-b) CP distribution is a Borel probability measure Q∈ P(X) that is T invariant and adapted (defined in the next paragraph). A (base-b) CP process is a measure preserving dynamical system (X,B, Q, T ) where B is the Borel σ-algebra on X and Q is a CP distribution. Probability measures on X orP(ΛN) are called distributions in order to distinguish them from measures on smaller spaces such as ΛN.The projection π1: X→ P(P(ΛN)) allows us to associate to the distribution Q∈ P(X) its measure marginal Q = (π1)∗Q∈ P(P(ΛN)). We define Q to be adapted if for all f∈ C(X), ZZZ f (µ, w) dQ(µ, w) =f (µ, w) dµ(w)dQ(µ). Adaptedness means “for Q-a.e. (µ, w)” is interchangeable with “for Q-a.e. µ, for µ-a.e. w.” Given P∈ P(P(ΛN)), there is a unique adapted distribution Q∈ P(X) for which Q = P ; if Q is adapted, then Q(X) = 1, so Q ∈ P(X). Since CP distributions are (by definition) adapted, it is common to speak of them as being supported onP(P(ΛN)) and to write suppQ to mean suppQ. 2858Oberwolfach Report 47/2017 The simplest examples of CP distributions are those supported on a single measure; it is an easy exercise to show that µ∈ P(ΛN) is a product measure if and only if there exists a CP distribution Q such that suppQ =µ. This example already demonstrates a basic connection between the T -invariance of Q and the fine-scale structure of measures in suppQ. As a related example, if µ, ν∈ P(ΛN) are such that for all λ∈ Λ, µλ= ν and νλ= µ, then the adapted distribution with measure marginal (δµ+ δν)/2 is a CP distribution. There are two other examples of CP distributions that we will just mention here. For a σ-invariant measure µR∈ P(ΛN), the adapted distribution Q with Q = δδwdµ(w) is a CP distribution supported entirely on point masses. Furstenberg describes an extension of this example with prediction measures in [2, pg. 409]. 2. Micromeasure distributions and dimension Given a measure µ∈ P(ΛN) and a point w∈ suppµ, the trajectory of µ through P(ΛN) alluded to above is (µw1···wn)n∈N, the sequence of probability measures seen around w in µ. A micromeasure of µ is a limit point of such a trajectory. If µ has dynamical or combinatorial origins, its micromeasures can often be related back to itself. An example of this is given in the following lemma. Lemma 1. If µ∈ P(ΛN) is σ-invariant and ν is a micromeasure of µ, then suppν⊆ suppµ. The setnMD(µ, w) of micromeasure distributions of µ at w is the set of limit points of1nPn−1i=0δTi(µ,w)oinP(X). Micromeasure distributions are supn∈N ported on the micromeasures of µ. Just as in the theorem of Krylov and Bogolioubov, the setMD(µ, w) is non-empty by the compactness of P(X) and every element is T -invariant, provided it is supported on X. The following theorem says that most of the time, this caveat is satisfied. Theorem 1 ([5, Theorem 28]). For all µ∈ P(ΛN), for µ-a.e. w∈ ΛN, every element ofMD(µ, w) is a CP distribution. Thus micromeasure distributions provide a rich array of CP distributions. Two useful facts about ergodic CP distributions – those Q for which (X,B, Q, T ) is ergodic – follow quickly from Theorem 1: since almost every point is generic for an ergodic measure, if Q is ergodic, then for Q-a.e. (µ, w),MD(µ, w) = Q; and the ergodic components in the ergodic decomposition of a CP distribution are themselves CP distributions. The latter fact is useful as it allows us to concentrate on ergodic CP distribu-R tions. The dimension of an ergodic CP distribution Q is dim Q =H ν,C1dQ(ν), the Q-average Shannon entropy of measures inP(ΛN) with respect to the partition C1=[λ]| λ ∈ Λ. Measures which support ergodic CP distributions have nice dimensionality properties, as indicated in the following theorem. Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2859 Theorem 2 ([2, Theorem 2.1]). Let Q be an ergodic CP distribution. The Qtypical measure µ is exact dimensional with dim µ = dim Q: for µ-a.e. w∈ ΛN,  log µ[w1· · · wn] lim= dim Q. n→∞n Combining the ideas behind micromeasure distributions and Theorem 2, we can construct from a measure µ an ergodic CP process supported on the micromeasures of µ with dimension bounded from below. Theorem 3 ([4, Theorem 7.10]). Let µ∈ P(ΛN). There exists an ergodic CP distribution of dimension at least lim supn→∞H(µ,Cn)n, whereCn=[v]| v ∈ Λn, supported on the micromeasures of µ. More can be said about CP processes such as the ones arising in Theorem 3 than about the specific measures from which they arise; this will be, in part, the subject of the following talks. Some properties of these CP processes then pass back to the originating measure µ via results such as the one in Lemma 1. References
[18] H. Furstenberg, Intersections of Cantor sets and transversality of semigroups, in Problems in Analysis (Sympos. Salomon Bochner, R. C. Gunning (Ed.)), Princeton Univ. Press, 50:7040 (1970), 41–59
[19] H. Furstenberg, Ergodic fractal measures and dimension conservation, Ergodic Theory Dynam. Systems, 28 (2008), 405–422 · Zbl 1154.37322
[20] M. Hochman, Lectures on dynamics, fractal geometry, and metric number theory, Journal of Modern Dynamics 8 (2015), 437–497 · Zbl 1314.28008
[21] M. Hochman, P. Shmerkin, Local entropy averages, Ann. Math 175 (2012), 1001–1059 · Zbl 1251.28008
[22] P. Shmerkin, Ergodic geometric measure theory, Unpublished, available upon request (2012)
[23] M. Wu, A proof of Furstenberg’s conjecture on the intersections of ×p and ×q-invariant sets, ArXiv e-prints, https://arxiv.org/abs/1609.08053 Furstenberg’s Dimension Conservation Theorem and Local Entropy Averages Simion Filip This talk developed results on CP processes due to Furstenberg [Fur08] and Hochman & Shmerkin [HS12]. The first basic result is concerned with projections of measures on trees. Suppose that X, Y are two trees and π : X× Y → X is a projection onto the first factor. A measure θ on X× Y yields the pushed-forward measure π∗θ on X and also conditional measures θxfor a.e. x∈ X. Furstenberg’s result [Fur08, §3] shows that for random measures coming from CP processes the dimension of the projected measure and the dimension of the conditional measures add up to the total dimension of the measure θ. 2860Oberwolfach Report 47/2017 Theorem [Furstenberg Dimension Conservation] For an ergodic CP process, for a.e. measure θ the projection π∗θ is exact-dimensional, as are the fiberwise conditional measure θx, and we have dim θ = dim π∗θ + dim θx The proof of the theorem involves the following steps.First, for any CP processes there is an observable (the entropy for the first level partition) whose Birkhoff averages give the dimension of the measure. An adaptation of this construction, using fiberwise entropy, gives a formula for dim θxin terms of an expression resembling, but not quite equaling a Birkhoff sum. Untangling the expression and applying a variant of the Birkhoff ergodic theorem implies the end result. Furstenberg used his result to show that dimension conservation holds for selfsimilar fractals A⊂ Rm1+m2for a projection π : Rm1+m2→ Rm1. Namely, starting from a self-similar fractal he builds a CP process, adapted to the projection in question. Applying the theorem above implies dimension conservation in the following sense: there exists δ > 0 such that  δ + dimx∈ Rm1: dim π−1(x)≤ δ≥ dim A By convention, the dimension of the empty set is−∞. The next result discussed is due to Hochman & Shmerkin [HS12,§4] and is a key tool in proving further results on agreement of expected and actual dimension in later talks. The Birkhoff sums used to compute dimension of measures for CP processes are now replaced by local entropy averages. The advantage of entropy averages is that they can be estimated in terms of local quantities. The result applies to any measure on a tree, not just one coming from a CP process. For a point x in a tree X, which we view as a point at infinity in the tree, denote by [xn1] the level n cylinder containing X. Then we have: Theorem [Local Entropy Averages] Suppose that µ is a measure on a tree X and that for µ-a.e. x∈ X we have lim inf− log µ([xn1])≥ α n→∞n Then we have dimµ≥ α. There is also a relative version of this theorem which is useful when estimating dimensions of projections. To describe it, denote for x∈ X and level n cylinder set [xn1] the conditional measure µ[xn1]induced on the symbol set by µ([xn1λ]) µ[xn](λ) := 1µ([xn]) 1 Theorem [Local Entropy Averages, Relative Version] For a morphism of trees f : X→ Y and a probability measure µ on X, suppose that N −1 1X NH(f∗(µ[xn1]))≥ α k=0 for µ-a.e. x. Then we have dimf∗µ≥ α. Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2861 The proof of this theorem is by finding a “random” section σ : Y→ X of the projection map f : X→ Y . The sections are chosen uniformly at random in each fiber and one applies the previous result to the measure σ∗f∗µ; the lower bound for the dimension of f∗µ then follows by averaging. References [Fur08] FurstenbergH.—“Ergodicfractalmeasuresanddimensionconservation”.ErgodicTheoryDynam.Systems28no.2,(2008)405–422. http://dx.doi.org/10.1017/S0143385708000084. [HS12] HochmanM.&ShmerkinP.—“Localentropyaveragesandprojections of fractal measures”. Ann. of Math. (2) 175 no. 3, (2012) 1001–1059. http://dx.doi.org/10.4007/annals.2012.175.3.1. Hochman–Shmerkin projection theorems Laurent Dufloux This talk is based on [1]; we restrict ourselves to dimension 2 in order to simplify the exposition, and we skip the part of this paper which deals with products of Gibbs measures. We first state Hochman–Shmerkin projection theorem. This result is then applied to self-similar measures with dense rotations, and products of×2 and ×3 invariant measures, settling a conjecture of Furstenberg. 1. Projection theorem In previous talks, CP-distributions were defined in trees; the definition of a CPdistribution in Euclidean spaces is essentially the same, with nested dyadic partitions playing the role of nested cylinders. See [1] for a more general definition. If P is a CP-distribution in R2and π is orthogonal projection from R2onto some line of R2, we let Z EP(π) =dP (µ) dim(π∗µ). Theorem 1 ([1] Theorems 8.1 and 8.2). Let P be an ergodic CP-distribution of dimension α∈ [0, 2]. (1) For almost every π, EP(π) = infα, 1. (2) If π is fixed, EP(π) = dim(π∗µ) for P -almost every µ. (3) For P -almost every µ, dim(π∗µ)≥ EP(µ) for every π. (4) The mapping π7→ EP(π) is lower semi-continuous. The proof of this result relies on local entropy averages bounds and Marstrand’s projection theorem (for the first statement). The main point is that local entropy averages bounds, along with the (statistical) “self-similarity” property of a random µ, allow to consider the entropy of projected measures at a fixed scale, and this is then “essentially” a continuous function of π. 2862Oberwolfach Report 47/2017 2. Projections of self-similar measures Consider an IFSfi; i∈ Λ where Λ is finite, and the fiare contracting similarities of R2. We assume that this IFS satisfies the strong separation condition, i.e. is X is the attractor of the IFS, the fPi(X) are pairwise disjoint. Let µ be a self-similar measure, i.e. µ =ipi(fi)∗µ, where (pi) is a probability vector with strictly positive components. Theorem 2 ([1], Theorem 1.6). Assume that the rotation parts of the figenerate a dense semigroup of SO(2). Then for any linear projection π from R2onto R, dim(π∗µ) = inf1, dim(µ). Proof. It is possible to construct an ergodic CP-distribution P satisfying the property that for P -almost every ν, there is an affine similarity S such that µ is absolutely continuous with respect to S∗ν. An application of Theorem 1 then shows that, given ε > 0, the set of projections π such that π∗µ has dimension at least inf1, dim(µ) − ε is dense and open, and the hypothesis on the rotation parts of the fithen implies that actually dim(π∗µ) > inf1, dim(µ) − ε for every π. The corresponding result for projections of self-similar sets follows from the theorem for self-similar measures. 3. Furstenberg’s conjecture The following result was stated, but not proved in the course of the talk: Theorem 3 ([1], Theorem 1.3). Let µ (resp. ν) be a×2 (resp ×3) invariant measure on [0, 1]. Let θ be the product measure θ = µ⊗ ν. Then for every orthogonal projection π which is not one of the coordinate projections, dim(π∗θ) = inf1, dim(θ). This is a strengthening of a conjecture of Furstenberg dealing with sets rather than measures. The statement for sets follows from the statement for measures, using the variational principle. The proof is quite technical. It relies on Theorem 1 (more precisely, a version of this result for non-ergodic CP-distributions) and the construction of a “generalized” CP-distribution, where dyadic partitions are replaced with a family of partitions by rectangles of bounded eccentricity. The measure θ is invariant by the product transformation (×2, ×3) which is non-conformal; in order to be able to zoom in on θ in a meaningful way, one is led to construct a dynamical system living above an irrationnal rotation, and the zooming process is a skew product over this rotation. References
[24] Michael Hochman and Pablo Shmerkin. Local entropy averages and projections of fractal measures. Ann. of Math. (2), 175(3):1001–1059, 2012. Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2863 Wu’s Proof of Furstenberg’s Intersection Conjecture Tom Kempton This talk was based on recent work of Wu [1], in which he proved the following theorem, originally conjectured by Furstenberg. Theorem 1. If A, B⊂ [0, 1] are closed and invariant under ×p, ×q respectively, and iflog plog q∈ Q, then for all real numbers u and v,/ dimH((uA + v)∩ B) ≤ max0, dimH(A) + dimH(B)− 1. We focused on the special case that A is the middle-13Cantor set and B the middle-12Cantor set, this is notationally simpler and allows for good pictures to be drawn, but is actually not much easier than the proof of the full theorem. The sets (uA + v)∩ B can be thought of (up to an affine coordinate change) as slices through the product set A× B. Wu’s proof involves showing that that, if there is a slice through A× B of upper box dimension γ, then (1) For Lebesgue almost every θ there exists a slice lθthrough A× B with slope θ and dimH(lθ∩ (A × B)) ≥ γ. (2) Furthermore, these slices lθcan be chosen such that there is a set C⊂ A×B of small box dimension such that each lθintersects B. (The real statement is a little more complicated, but follows this idea) (3) Putting 1 and 2 together gives that A× B must have dimension at least 1 + γ. Part 1 was originally proved by Furstenberg. The proof involves building CP chains supported on slices through A× B of Hausdorff dimension at lest γ. The majority of the talk was spent proving part 1 and showing how parts 1 and 2 together are enough to show part 3. References
[25] M. Wu, A proof of Furstenberg’s conjecture on the intersections of ×p and ×q-invariant sets, (arXiv: 1609.08053) Some additive combinatorics Thomas F. Bloom 1. Introduction to additive combinatorics I give an introduction to some basic tools and concepts of additive combinatorics, in their traditional context of set addition and in terms of entropy, the latter following in particualar the paper of Tao [2]. Let G be an abelian group, which for convenience I will take to be finite, and let A, B⊂ G. The sumset A + B is defined as A + B =a + b : a ∈ A, b ∈ B. In this talk I will discuss inequalities between sizes of sumsets, and also what kind of structural information can be deduced from knowing that such sizes are small. 2864Oberwolfach Report 47/2017 (1) Inequalities from trivial identities: There is a very useful family of relationships between the sizes of sumsets, of which the most useful is the following sumset inequality due to Ruzsa, often known as Ruzsa’s triangle inequality: |A − C||B| ≤ |A − B||B − C|, which follows since the trivial identity (a− b) + (b − c) = a − c implies that the map (a− c, b) 7→ (a − b, b − c) is an injection (choosing a unique representative for each a− c ∈ A − C). (2) Covering lemmas: Often the starting point for the proof of inverse theorems discussed below, these show that sets with small sumset can be efficiently covered by a small number of translates. Again, the classic example is due to Ruzsa: if|A + B| ≤ K|B| then A is contained in the union of at most K translates of B− B. The proof is just to take a set X⊂ A maximal such that the translates x + B are pairwise disjoint for all x∈ X. (3) Pl¨unnecke’s inequality: Suppose that|A + A| is small compared to |A|. Since set addition is a smoothing operation, one would hope that this property is preserved under more additions, e.g.|A + A + A| continues to be small, and so on. This is made rigorous by the following inequality of Pl¨unnecke: if|A + A| ≤ K|A| then for all t ≥ 2 |tA| ≤ Kt|A|, where tA = A +· · · + A is the t-fold sumset of A. (4) Inverse theorems: What structural information about A can we deduce if|A + A| ≪ |A|? Such an inequality clearly holds if A is dense in some multidimensional arithmetic progression. The Freiman-Ruzsa inverse theorems show that this is the only possibility, allowing us to deduce strong algebraic structure from quite weak statistical information. We will go into the latter two topics in more depth shortly. Two other aspects of additive combinatorics important for this workshop are the Balog-Szemer´ediGowers theorem and the sum-product phenomenon, which will be discussed in separate talks. 2. Entropy analogues Let X be a random variable taking values in G. If X is uniformly sampled from some A⊂ G then the entropy H(X) is exactly log|A|. This leads us to ask similar questions as in the previous section, but now considering arbitrary random variables. For example, what can we deduce about X if H(X + X)− H(X) is small? Are there sumset inequalities analaogous to the Ruzsa triangle inequality? It is important to note an important distinction between the combinatorial and entropy worlds – even if X is sampled uniformly from A, X + X is not necessarily uniformly sampled from A + A. Instead, the corresponding probability measure is proportional to 1A∗ 1A, which is much smoother than 1A+A. Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2865 For example, consider the set A which is the union of two arithmetic progressions in Z2along the orthogonal axes. Sampling uniformly from A + A would almost surely give a point inside the integer lattice on a box, while sampling according to 1A∗ 1Awould, with positive probability, give a point on the axes. While the relationship between cardinalities of sumsets and entropy of sums of random variables is close, neither can be deduced from the other in general, and both must be developed in parallel. The important facts about entropy H(X) we need are that it is a non-negative quantity, however we condition on other random variables, that conditioning decreases entropy, H(X|Y ) ≤ H(X), and we have the chain rule H(X|Y ) + H(Y ) = H(X, Y ). From this it is straightforward to deduce the submodularity inequality: if X and Y both individually determine Z and the joint distribution (X, Y ) determines W then H(Z) + H(W )≤ H(X) + H(Y ). This has two important consequences. (1) Ruzsa triangle inequality: Let X, Y, Z be independent random variables. Since (X− Y, Y − Z) and (X, Z) each independently determine X− Z and (X, X − Y, Z, Y − Z) determines (X, Y, Z), H(X− Z) + H(X, Y, Z) ≤ H(X − Y, Y − Z) + H(X, Z) whence by independence H(X− Z) + H(Y ) ≤ H(X − Y ) + H(Y − Z), the entropy analogue of Ruzsa’s triangle inequality. (2) Kaimanovich-Vershik inequality: This is an entropy analogue of Pl¨unnecke’s inequality. Let X, Y, Z be independent random variables. Since (X+Y, Z) and (X, Y +Z) each determine X+Y +Z and (X+Y, Z, X, Y +Z) determines (X, Y, Z), the submodularity inequality yields, after rearrangement, H(X + Y + Z)− H(X + Z) ≤ H(X + Y ) − H(X). In particular, if H(X + Y )≤ H(X) + log K then, if Y1, . . . , Ytare independently sampled copies of Y , then iterating the above gives H(X + Y1+· · · + Yt)≤ H(X) + t log K. This inequality will be useful many times in the subsequent talks. 3. Pl¨unnecke’s inequality The direct analogue of the previous entropy inequality for cardinalities would be that, if A, B, C are any finite sets, then |A + B + C||A| ≤ |A + B||A + C|. This inequality is false, however, since the cardinality of a sumset is much less robust than the entropy of the sum of random variables. Petridis observed that if we are able to pass to some subset A′⊂ A, however, conditioned only on B, then 2866Oberwolfach Report 47/2017 this becomes true. That is, for any sets A, B there is A′⊂ A (depending on B) such that for all sets C |A′+ B + C||A| ≤ |A + B||A′+ C|. Petridis gave a very elegant proof of this fact, using only elementary combinatorics. Pl¨unnecke’s inequality is a simple deduction of this inequality. 4. Inverse theorems How can|A + A| be small compared to |A|? For convenience, we will now assume that G = Z. It is easy to check that d-dimensional arithmetic progressions are an example: a progression of rank d is a set of the shape P =a0+ a1n1+· · · + adnd: 0≤ ni< Ni for some integers ai, Ni. Whatever the choice of parameters,|P + P | ≤ 2d|P |. Furthermore, if A is any large subset of P then it must also have small doubling for trivial reasons. A important and deep result of Freiman and Ruzsa is that the converse also holds. That is,|A + A| ≤ K|A| if and only if there is a progression P of rank d≪K1 such that A⊂ P and |A| ≫K|P |. Tao proved the following natural entropy analogue: if H(X+X)≤ H(X)+log K then there is a progression P of rank d≪K1 such that X is close in transport distance to µP, the uniform measure on P . That is, there is some Z (possibly dependent on X) such that H(Z)≪K1 and X + Z≡ µP. The proof first reduces to the case when X is close to the uniform measure on some set A, and then invokes the previous Freiman-Ruzsa inverse result. References
[26] G. Petridis, New proofs of Pl¨unnecke-type estimates for product sets in groups, Combinatorica, 32 (2012), 721–733. · Zbl 1291.11127
[27] T. Tao, Sumset and inverse sumset theory for Shannon entropy, Combin. Probab. Comput. 19(2010) 603–639. · Zbl 1239.11015
[28] T. Tao and V. Vu, Additive combinatorics, Cambridge University Press (2006). Hochman’s inverse theorem on the growth of entropy under convolutions Mikolaj Fraczyk Hochman’s inverse theorem, introduced in [1], describes the multiscale structure of probability measures µ, ν on [0, 1) for which scale 2−n−entropy of the convolution µ∗ ν is very close to scale 2−n−entropy of µ. To make the statement precise we define below the scale 2−n−entropy as well as other necessary notions. Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2867 0.1. Notations. 0.1.1. Dyadic partitions. Let I be an interval in R. WriteP(I) for the space of probability measures on I. For n∈ Z we the partition Dnof R is given as Gkk + 1 2n,2n. k∈Z If x∈ R we write Dn(x) for the unique cell in Dncontaining x. For a measure µ∈ P(R) and a cell D ∈ Dmsuch that µ(D)6= 0) we write 1 µ(D)µ|Dand µD:= (TD)∗µD where TD: D→ [0, 1) is the unique bijective homothety between D and [0, 1). Measure µDis called the raw D-component of µ and µDis the rescaled Dcomponent of µ. For x∈ R we adopt convention µx,i= µDi(x)and µx,i= µDi(x). 0.1.2. Entropy, almost atomic and almost uniform measures. Let µ∈ P(R). The normalized entropy of µ at scale 2−nis defined as Z 11 nnR− log(µ(Dm(x))dx. Let ε > 0, m≥ 0. We say that measure µ ∈ P([0, 1)) is (ε, m)-atomic if Hm(µ)≤ ε and (ε, m)-uniform if Hm(µ)≥ 1−ε. Intuitively, when ε is small the measure µ is (ε, m)-atomic is its mass in concentrated in a single cell of Dmand (ε, m)-atomic if its mass is almost uniformly distributed on the cells of Dmin [0, 1). 0.1.3. Probability and expected value. We adopt following conventions: Let A⊂ P([0, 1)) be event and let f : P(R) → R a measurable function. For measures µ, ν∈ P(R) we write P(µx,i∈ A) := µ(x ∈ R | µx,i∈ A) andZ E(f (µx,i)) :=f (µx,i)dµ(x). When two or more measures are involved we will treat their component measures as independent random variables. In particular Z E(f (µx,i∗ νy,j)) :=f (µx,i∗ νj,y)dµ(x)dν(y). 0.2. Inverse theorem. As we explained in the first paragraph Hochman’s inverse theorem describes the structure of measures µ, ν on [0, 1) such that for n-big and δ very small we have Hn(µ∗ ν) ≤ Hn(µ) + δ. Note that the inequality is satisfied trivially if µ is uniform (i.e. Lebesgue on [0, 1)) and ν is supported on a small ball or when ν is atomic and µ is any probability measure. Let us give a more complicated example. 2868Oberwolfach Report 47/2017 0.2.1. Motivating example. Fix δ > 0 and a natural number n, it has to be big when δ is very small, say n >> δ−1. We will construct a probability measure µ on [0, 1) with the property that Hn(µ∗ µ) ≤ Hn(µ) + δ. Choose k≤ δn and choose natural numbers 1 = a1< b1< a2< b2< . . . < ak< bk= n. Consider the set Σ of rational numbers of form q =2mn, 0≤ m < 2nsuch that q has non-zero binary digits only on positions in the intervals [aPi, bi) for i = 1, . . . k. We choose the numbers ai, biin such a way that N :=ki=1(bi− ai) (i.e the number of positions where the digit in not fixed) is roughly of size n/2. The measure µ is defined as the normalized counting measure on Σ. The cardinality of Σ is 2Nand the atoms are separated by at least 2−nso we can compute the scale 2−n−entropy as follows Hn(µ) =n1H(µ) =1nlog 2N=Nn∼12. To estimate the entropy of the convolution µ∗ µ we will simply bound the cardinality of the support of µ ∗ µ. Let x, y∈ Σ, we claim than the sum x + y can have non zero binary digits only on places in the intervals [ai− 1, bi) for i = 1, . . . , k. Indeed, if the m− th digit of x + y is non-zero then either m−th of m + 1-th digit of one of x or y had to be non-zero. Hence, the non-zero digits of numbers in Σ can appear only on places inSki=1[ai− 1, bi) which implies a bound|Σ + Σ| ≤ 2N +k. It follows that Hn(µ∗ µ) =n1H(µ)≤N +kn= Hn(µ) +Nk. Recall that k was chosen so that k≤ δN so we have Hn(µ∗ µ) ≤ Hn(µ) + δ. We end this paragraph with the multiscale analysis of µ. Choose m small relatively to n. We are interested in the scale−2−mstructure of the rescaled component measures µx,j, j = 1 . . . , n. It turns out that when the scale j is inside [ai, bi− m) then the rescaled components µx,iare, at scale 2−mas uniform as possible i.e. Hm(µx,i) = 1. On the other hand when the scale j is in [bi, ai+1− m) then the measures µx,iare atomic at scale 2−mi.e. Hm(µx,i) = 0. The set of scale where this dychotomy doesn’t hold is contained in the union Sk i=1([ai−m, ai)∪ [bi− m, bi)) so its cardinality is roughly of size 2km≤ 2δnm. We see that for m small compared to n the dychotomy between uniformness and atomicity holds at most scales between 1 and n. Inverse theorem says that when we relax a bit the notions of uniformness and atomicity, such a dychotomy holds at almost all scales whenever Hn(µ∗ µ) ≤ Hn(µ) + δ. 0.2.2. Statement of the theorem. Theorem 1 (Theorem 2.7 and Theorem 4.11 [1]). For every ε > 0 and integer m≥ 1, there is a δ = δ(ε, m) > 0 such that for every n > n(ε, δ, m), the following holds. If µ, ν∈ P([0, 1)) and Hn(µ∗ ν) < Hn(µ) + δ, then there are disjoint subsets I, J⊂ 1, . . . , n with |I ∪ J| > (1 − ε)n, such that  P µx,kis (ε, m)− uniform> 1− ε for k ∈ I,  P νx,kis (ε, m)− atomic> 1− ε for k ∈ J. Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2869 References
[29] M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Annals of Mathematics 180 (2014), 773–822. Hochman’s Theorem on self similar measures with overlap Amir Algom The purpose of my talk was to discuss Hochman’s method (see [1]) of computing dimensions of self similar measures and sets using the inverse theorem for entropy, proved in the previous lecture. Let Φ =φii∈Λ,|Λ| ≥ 2 be a finite family of real linear contractions; that is, for every i∈ Λ the map φi: R→ R is defined by φi(x) = ri· x + aiwhere|ri| < 1 and ai∈ R (recall that Φ is called an iterated function system - IFS). To avoid trivialities we assume throughout that there are at least two distinct contractions. Let X denote the attractor of Φ (the existence of such a set was already established in a previous talk). Let µ be the self-similar measure associated with Φ and a nondegenerate probability vector (pi)i∈Λ. For a Borel probability measure θ on R we denote dim θ = infdim A : θ(A) > 0. This notion is sometimes known as lower-Hausdorff dimension. There are other notions of dimension, but for self similar measures, that are exact dimensional, most major ones coincide. The aim of this talk was to discuss Hochman’s approach to computing the dimension of the self similar measure µ. The classical approach to dimension theory of self similar measures is to impose some separation condition on Φ (e.g. the strong separation condition), and deduce that dim X = s− dim X (the latter denotes similarity dimension). Similarly (assuming again some separation of Φ), dim µ = s−dim µ (the latter denotes the similarity dimension of the self similar measure µ, introduced in a previous talk). It is when the images φi(X) have significant overlap that computing the dimension becomes difficult, and much less is known. The main strength of Hochman’s approach is its ability to yield non trivial information about the small scale geometry of the measure µ in this situation. Notation For i = i1...in∈ Λnwrite • φi= φi1◦ ... ◦ φin, and call this a cylinder map. • ri= ri1· · · rin, the contraction ratio of φi. • Similarly, we given a proabability vector (pi)i∈Λwe write pi= pi1· · · pin. Let n∈ N, and fix i 6= j ∈ Λn. We define the distance between the ngenerational cylinders i, j by dn(i, j) =|φi(0)− φj(0)|if ri= rj and otherwise denote dn(i, j) =∞. We define ∆n= mind(i, j) : i 6= j ∈ Λn. 2870Oberwolfach Report 47/2017 We note the following observations: • The definition is unchanged if we pick any other point in R. • The are exact overlaps in Φ if and only if ∆n= 0 for some n. • ∆n→ 0 exponentially. • There can be an exponential lower bound for ∆n: this happens if the images φi(X) are disjoint, under the OSC, or for example when the parameters of Φ are algebraic. The main result on self-similar measures, presented in my talk, was the following: Theorem 1. If µ is a self similar measure on R and dim µ < min(1, s− dim µ) then ∆n→ 0 super-exponentially, i.e. limn− log ∆nn=∞ The conclusion is about ∆n, which is determined by the IFS, not by the measure. Thus, if the conclusion fails, then dim µ = s− dim µ for every self-similar measure of Φ. Also, the same statement remains true for the attractor X, i.e. if dim X < min(1, s− dim X) then ∆n→ 0 super-exponentially. Theorem 1 is derived from a more quantitative result about the entropy of finite approximations of µ, which we now describe. Recall • We write H(µ, E) for the Shannon entropy of a measure µ with respect to a partition E, and H(µ, E|F ) for the conditional entropy on F . • For n ∈ Z the dyadic partitions of R into intervals of length 2−nis kk + 1 2n,2n) : k∈ Z
[30] M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Annals of Mathematics. Second Series 180 (2014), 773–822. Background on Bernoulli convolutions Korn´elia H´era P∞ For λ∈ (0, 1), let νλbe the distribution ofn=0±λn, where the signs are chosen independently with probability12. It can be written as the infinite convolution product νλ=∗∞n=012(δ−λn+ δλn), hence the measures νλare called Bernoulli convolutions (BC). They have been studied since the 1930’s, revealing surprising connections with harmonic analysis, the theory of algebraic numbers, dynamical systems, and Hausdorff dimension estimation. In this talk we touched on some of the main classical and modern results about BC, based on [7]. Jessen and Wintner (1935, [4]) showed that νλis either absolutely continuous, or purely singular, depending on λ. Kershner and Wintner (1935, [6]) observed that νλis singular for λ∈ (0,12), since it is supported on a Cantor set of zero Lebesgue measure, and ν1is uniform on [−2, 2]. The main question about BC is 2 the following: Question 1. For which parameters λ∈ (12, 1) is νλabsolutely continuous? If absolute continuity holds, what can we say about the density? 2872Oberwolfach Report 47/2017 The following properties of BC are useful to understand their structure better. We will use the notation f µ for the push-forward of µ by f : f µ = µ◦ f−1. (1) νλcan be characterized as the unique probability measure satisfying 11 νλ=S−1νλ+S1νλ, 22 where Si(x) = λx + i for i = 1,−1. Thus νλis a self similar measure for the IFSS−1, S1 with weights12. (2) Let Ω =−1, 1N, µ = (12,12)Nthe Bernoulli measure on Ω, and X∞ πλ: Ω→ R, ω →ωnλn. n=0 Then νλ= πλµ. This point of view is useful when ideas from geometric measure theory are applied, in particular in the transversality method. (3) The Fourier transform of a finite Borel measure ν on R is defined as ˆν(u) = R e−2πiuxdν(x). Easy computation gives Y∞ νˆλ(u) =cos(2πλnu). n=0 The formula is important when using methods related to number theory. The first answer to Question 1 was given by Erd˝os in 1939. Definition 1. β > 1 is a Pisot number if it is an algebraic integer such that all other roots of its minimal polynomial have modulus less than 1. Theorem 1 (Erd˝os, 1939, [1]). If λ∈ (0, 1) 12, β =λ1is a Pisot number then νλis singular. The next theorem of Erd˝os revealed that near 1, absolute continuity is generic. Theorem 2 (Erd˝os, 1940, [2]). There exists ε > 0 such that for almost all λ∈ (1− ε, 1), νλis absolutely continuous. However, explicit bounds for the neighborhood of 1 were not given. Kahane (1971, [5]) gave a brief outline of the argument and indicated that dimλ ∈ (1 − ε, 1) : νλis singular → 0 as ε → 0. Here and in the sequel dim always denotes Hausdorff dimension. The following theorem has come to be known as the ”Erd˝os-Kahane argument”. √ Theorem 3. Let a =2, b = 2, k≥ 3 be an arbitrary integer. Then dimλ ∈ [b−1, a−1] : ˆνλ(u)6= O(u−0.02/k) ≤ 3log(3000k). k The theorem can be formulated for arbitrary 1 < a < b <∞ with appropriate constants in the corresponding places. The proof is combinatorial in nature, an exposition of the argument can be found in [7]. Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2873 Denote by P the family of measures on R which have a power Fourier decay at infinity, that is, there exist σ, C > 0 such that|ˆµ(u)| ≤ C|u|−σfor all u∈ R. Then theorem 3 easily implies: Corollary 1. dimλ ∈ (0, 1) : νλ∈ P = 0./ The following corollary can be derived from Theorem 3 using the convolution structure of νλ, namely that ˆνλ(u) = ˆνλ2(u)· ˆνλ2(λu). Corollary 2. For any s > 0 and m∈ N, there exists ε > 0 such that dνλ dimλ ∈ (1 − ε, 1) : the density∈ C/m < s. dx Garsia (1962, [3]) found an explicit set of parameters for which absolute continuity holds. Definition 2. β > 1 is a Garsia number if it is an algebraic integer such that all roots of its minimal polynomial have modulus greater than 1, and the minimal polynomial has constant term±2. Theorem 4 (Garsia, 1962, [3]). If λ∈ (12, 1), β =1λis a Garsia number then νλ is absolutely continuous. After Erd˝os’s result from 1940 on generic absolute continuity near 1, an obvious question was whether νλis absolutely continuous for almost all λ in the maximal possible interval (12, 1). Solomyak [9] gave an affirmative answer in 1995, moreover showing that νλhas an L2density for almost all λ∈ (12, 1). Soon after, Peres and Solomyak [8] gave a simplified proof which avoided use of the Fourier transform. Both proofs rely on the so-called transversality method. Theorem 5 (Solomyak, 1995, [9]). For almost all λ∈ (12, 1), νλis absolutely continuous with an L2density. In this talk we gave a brief outline of the proof from [8], by showing the basic ideas of transversality. References
[31] P. Erd˝os, On a family of symmetric Bernoulli convolutions, Amer. J. Math. 61 (1939), 974–975.
[32] P. Erd˝os, On the smoothness properties of Bernoulli convolutions, Amer. J. Math. 62 (1940), 180–186.
[33] A. M. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102(1962), 409–432. · Zbl 0103.36502
[34] B. Jessen, A. Wintner, Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc. 38 (1935), 48–88.
[35] J. P. Kahane, Sur la distribution de certaines series aleatoires, Colloque Th. Nombres [1969, Bordeaux], Bull. Soc. math. France, Memoire textbf25 (1971), 119–122.
[36] R. Kershner, A. Wintner, On symmetric Bernoulli convolutions, Amer. J. Math. 57 (1935), 541–548. · Zbl 0012.06302
[37] Y. Peres, W. Schlag, and B. Solomyak, Sixty years of Bernoulli convolutions, In Fractal geometry and stochastics, II (Greifswald/Koserow, 1998), volume 46 of Progr. Probab., pages 39–65. Birkh¨auser, Basel, 2000. 2874Oberwolfach Report 47/2017 · Zbl 0961.42006
[38] Y. Peres, B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Math. Res. Lett., 3(2) (1996), 231–239.P · Zbl 0867.28001
[39] B. Solomyak. On the random series±λn(an Erd¨os problem), Ann. of Math. 142 (1995), 611–625. Algebraic properties of Pisot and Salem numbers Mark Pollicot 1. Definitions: Pisot and Salem numbers and the Mahler measure Recall that an algebraic number β > 1 is a (real) root of a polynomial with integer coefficients P (z) := adzd+ ad−1zd−1+· · · + a1z + a0. with a0,· · · , ad∈ Z with ad, a06= 0. An algebraic integer β is a root of a (minimal) polynomial with integer coefficients, where the leading coefficient is unity, i.e., ad= 1. If β > 1 is an algebraic integer then its conjugate roots are the other d− 1 roots α1,· · · , αd−1of the corresponding polynomial, i.e., d−1Y P (z) = (z− β)(z− αi). i=1 Definition 1. We say that λ is a Pisot number if its conjugate roots have modulus strictly smaller than unity, i.e.,|αi| < 1, for i = 1, · · · , d − 1. A weaker condition is to say that λ is a Salem number if the conjugate roots have modulus less than or equal to unity, i.e.,|αi| ≤ 1, for i = 1, · · · , d − 1. We can also define the Mahler measure of any polynomial P∈ C[z] (of degree d with leading coefficient ad). Definition 2. The Mahler measure of P (z) is given by Y m(P ) =|ad||αi| ∈ R+ |αi|>1 The Mahler measure is related to another value associated to P , called the√ height h(P ) = maxi|ai|, by m(P ) ≤ h(p)d + 1. Example 1. The Mahler measure for P0(x) = x10+x9−x7−x6−x5−x4−x3+x+1 takes the value m(P ) = 1.17628 . . .. Indeed, this is the smallest known Mahler measure of a (non-trivial) polynomial. 2. The Lehmer conjecture The best known conjecture on Mahler measures is due to Lehmer. Conjecture 1 (Lehmer Conjecture). There exists δ > 1 such that for any polynomial P∈ Z[x] we have m(P ) ≥ δ or m(P ) = 1 For trivial cases). Moreover, the Salem number in Example 2 is supposed to be the one for which m(P ) > 1 is least. Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2875 3. Bernoulli convolutions Given β > 1 we can write the Fourier transform of the Bernoulli convolution measure ν associated to 0 < λ = β−1< 1 as: Y∞ νbλ(t) =cos (λnt) t∈ R. n=0 Recall that Erd¨os showed in 1939 that if β is Pisot thenbνλ(t) does not tend to zero (and thus by the Riemann Lebesgue lemma νλis not absolutely continuous). For Salem numbers we have the following: Lemma 1 (bν(t) doesn’t decay polynomially). If λ is a Salem number then for bν(t) and any ǫ > 0 we can choose tkր +∞ with bν(tk)tǫkր +∞. 4. Garsia’s lemma We have the following bound (see [2]) Lemma 2. Let β > 1 be an algebraic integer. Let α1,· · · , αs−1be the conjugate roots. Let σ = #1 ≤ i ≤ s − 1 : |αi| = 1. Assume P (x) ∈ Z[x] is a polynomial and height M and degree at most d with P (α)6= 0 then Q |P (α)| ≥Q|αi|≥1||αi| − 1|d+1 i|>1|αi|Ms Theorem 1 (Garsia’s separation theorem). Let 1 < β < 2 be a Pisot number corresponding to a polynomial of degree d, say. There exists C = C(β, m) > 0 such that if XXn ǫjβ−j6=ǫ′jβ−j j=1j=1 for some ǫj, ǫ′j∈ −1, 1 then (ǫj− ǫ′j)β−j ≥ Cβ−n. 5. Garsia entropy Let 0 < λ < 1 be any real number. Let 0 < p−1< 1 and p1= 1− p−1Consider the finite convolutions (p−1δ−λ+ p1δλ)∗ (p−1δ−λ2+ p1δλ2)∗ · · · ∗ (p−1δ−λn+ p1δλn) X =(pi1· · · pin)δ(−1)i1λ+(−1)i2λ2+···+(−1)inλn i1,··· ,in∈−1,1 which is supported on (distinct) points x1, x2,· · · , xr(r≤ 2n) with weights q1, q2,· · · , qr, say. 2876Oberwolfach Report 47/2017 Definition 3 (Garsia Entropy (see [3])). We then define Xr Hn=−qjlog qj j=1 and HN Hλ=lim N →+∞N The limit exists by a subaditivity argument. Clearly Hλ≤ log λ. Let us restrict to the case p−1= p1=12for simplicity. Theorem 1 (Garsia). If β > 1 is a Pisot number then for β = λ−1we have Hβ< log λ. In fact stronger inequalities are possible for Pisot numbers and the entropy is related to dimension of νλ. 6. Mahler’s seperation lemma If P∈ C[z] is an complex polynomial of degree d with distinct roots then z1,· · · , zd then Mahler gave a lower bound on the seperation of roots √p sup|zi− zj| ≥3|D(P )| i6=jdd/2+1M (P )d−1 p where|D(P )| = |ad−1d|Qi<j|zi− zj|. Let β > 1 be an algebraic number. Definition 4. LetPddenote the set of polynomials of degree at most d all of whose coefficients are−1, 0, 1. This has the following corollary. Lemma 3 (see [1]). Let d≥ 9. Let η 6= η′be two algebraic numbers each of which is a root of a polynomial inPd. Then|η − η′| > 2n−4n. 7. Hochman’s Theorem We can consider the following theorem and question (see [4], Theorem 1.9 and Question 1.10). Theorem 2 (Hochman). dim νλ= 1 outside a set of λ of dimension 0. Furthermore, the exceptional parameters for which dim νλ< 1 are “nearly algebraic” in the sense that for every 0 < θ < 1 and all large enough d, there is a polynomial pd(x)∈ Pnsuch that pd(λ) < θd. Question 2 (Hochman). Does there exist a constant s > 0 such that for α, β that are roots of polynomials inPdeither α = β or|α − β| > sd? The Lemma 3 above at least gives some lower bound (with s =d14, dependent on d). Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2877 References
[40] Emmanuel Breuillard and Pter Varj´u. On the dimension of Bernoulli convolutions. Preprint, arXiv:1610.09154, 2016.
[41] Adriano M. Garsia. Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc., 102:409-432, 1962. · Zbl 0103.36502
[42] Adriano M. Garsia. Entropy and singularity of infnite convolutions. Pacific J. Math., 13:1159-1169, 1963. · Zbl 0126.14901
[43] Michael Hochman. On self-similar sets with overlaps and inverse theorems for entropy. Ann. of Math. (2), 180(2):773-822, 2014. · Zbl 1337.28015
[44] Rapha¨el Salem. Algebraic numbers and Fourier analysis. D. C. Heath and Co., Boston, Mass., 1963 Shmerkin’s theorem on smoothness of BC Ariel Rapaport 1. The main Theorem Given λ∈ (0, 1) denote by νλthe unbiased Bernoulli convolution corresponding to the parameter λ. Theorem 1 (P.Shmerkin, [1]). There exists E⊂ (12, 1), with dimHE = 0, such that νλis absolutely continuous for all λ∈ (12, 1) E. 2. Correlation dimension, energies, and the Fourier transform Denote byP(R) the collection of all compactly supported Borel probability measures on R. Given µ∈ P(R) the (lower) correlation dimension of µ is defined by, R logµ(B(x, r)) dµ(x) (1)dimcµ = lim inf. r↓0log r It always holds that dimcµ≤ dimHµ. For s≥ 0 the s-energy of µ is defined by, Z Zdµ(x) dµ(y) Isµ =. |x − y|s It is not hard to verify that, (2)dimcµ = sups ≥ 0 : Isµ <∞ . The Fourier transform of µ is defined by, Z bµ(ξ) =eiξxdµ(x) for ξ∈ R . It is well known that for each 0 < s < 1 there exists a constant c(s) > 0 such that, Z (3)Isµ = cs|ξ|s−1|bµ(ξ)|2dξ . 2878Oberwolfach Report 47/2017 3. Shmerkin’s smoothing lemma Denote byD(R) the class of measures in P(R) whose Fourier transform has at least power decay, i.e. D(R) = µ ∈ P(R) : ∃ C, t > 0 s.t. |bµ(ξ)| ≤ C|ξ|−t∀ ξ ∈ R . Lemma 1. Let ν∈ D(R) and µ ∈ P(R) with dimHµ = 1, then ν∗ µ is absolutely continuous. Proof. Since ν∈ D(R) there exists C > 0 and t ∈ (0,12) with|bν(ξ)| ≤ C|ξ|−tfor all ξ∈ R. For n ≥ 1 let, 1 En=x ∈ R : µ(B(x, r)) ≤ r1−tfor all r∈ (0,) . n From dimHµ = 1 it follows µ(∪n≥1En) = 1. For n≥ 1 with µ(En) > 0 set µn=µ(Eµ|En. It is not hard to show, directly from (1), that dim n)cµn≥ 1 − t. From this and (2) it follows I1−2tµ <∞. Now by (3), ZZ | \ν∗ µn(ξ)|2dξ =|bν(ξ)|2· |cµn(ξ)|2dξ Z C2|ξ|−2t· |cµn(ξ)|2dξ =C2· I c1−2t1−2tµn<∞, which shows that ν∗ µnis absolutely continuous. Since this is true for every large enough n≥ 1 the lemma follows. 4. Proof of the main theorem Proof of Theorem 1. Given λ∈ (12, 1) and k≥ 2 consider the IFS k−1X fλ,kw(x) = λkx +(−1)wjλj: w∈ 0, 1k−1, j=1 and let νλ,k∈ P(R) be with X νλ,k=2−k+1· fλ,kwνλ,k. w∈0,1k−1 Note that log 2k−11 log λk= (1−k) dimsνλ, where dimsis the similarity dimension. Also observe that (4)νλ= νλk∗ νλ,k, P which follows from the fact that νλkis the distribution ofj±λkjand νλ,kis the distribution ofPj∤k±λj. From Hochman’s theorem, on parametric families of measures, it follows that there exists a set Ek⊂(12, 1),with dimHEk=0,such that dimHνλ,k= min1, dimsνλ,k for all λ ∈ (12, 1) Ek. Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2879 By the Erdos-Kahane argument there exists a set F⊂ (0, 1), with dimHF = 0, such that νλ∈ D(R) for all λ ∈ (0, 1) F . Set 1 2, 1) : λk∈ F
[45] Pablo Shmerkin. On the exceptional set for absolute continuity of Bernoulli convolutions. Geom. Funct. Anal., 24(3):946958, 2014. · Zbl 1305.28012
[46] Pablo Shmerkin and Boris Solomyak. Absolute continuity of self-similar measures, their projections and convolutions. Trans. Amer. Math. Soc., 368:51255151, 2016.
[47] Fedor Nazarov, Yuval Peres, and Pablo Shmerkin. Convolutions of Cantor measures without resonance. Israel J. Math., 187:93116, 2012. Varj´u’s theorem on smoothness of BC for algebraic parameters Weikun He This talk is based on the recent paper [3] due to Varj´u. In this talk, log denotes the logarithm of base 2. Let 0 < λ < 1. Denote by µλthe Bernoulli convolution with parameter λ. We have seen in a previous talk that for almost all λ∈ (1/2, 1), the BC µλis absolutely continuous. However, showing absolute continuity for explicit parameter λ is a different problem. Here we consider algebraic parameters. For an algebraic number λ, its Mahler measure Mλis defined to be the Mahler measure of its minimal polynomial. Previously, Garsia showed that µλis absolutely continuous if λ−1is an algebraic integer of Mahler measure 2. The main result of this talk is the following. Theorem 1. For any ǫ > 0 there is c > 0 such that the following is true. Let 0 < λ < 1 be an algebraic number. If 1− c min(log Mλ, (log Mλ)−1−ǫ) < λ < 1, then µλis absolutely continuous with respect to the Lebesgue measure. Note that in [3], Varj´u showed Theorem 1 for biased Bernoulli convolution µλ,p (the constant c then depends on ǫ and p). Moreover, if we restrict λ to the set of algebraic numbers which are not roots of any polynomial with coefficients in −1, 0, 1, the condition on λ can be replaced by an entirely explicit bound 1− 10−37(log(Mλ+ 1))−1(log log(Mλ+ 2)−3) < λ < 1, Thus, Theorem 1 gives new explicit examples of λ for which µλis absolutely continuous. Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2881 We outline the proof of Theorem 1. The main tool used in the proof is an averaged entropy at certain scale. Let X be a real valued random variable. Let r > 0 a real number. We define Z1jXk H(X; r) =H+ tdt 0r where H(· ) denotes the Shannon entropy. Moreover, for 0 < r < r′, define H(X; r| r′) = H(X; r)− H(X; r′). This notion of entropy enjoys several nice properties among which is the following Garsia’s absolute continuity criterion. A random variable is absolutely continuous with respect to the Lebesgue measure with class L log L density if and only if lim suplog(1/r)− H(X; r) < +∞. r→0+ In light of this, the proof of Theorem 1 reduces to the proof of (1)1− H(µλ; 2−n| 2−n+1)≤1 n2 for n≥ 1 sufficiently large under the assumption of Theorem 1. For an interval I⊂ (0, 1], we write µIλfor the distribution of the random variable X ξnλn n:λn∈I where (ξn) is a sequence of independent and identically distributed Bernoulli random variables taking value in−1, 1 with equal probability. With this notation, we have µλ= µIλ1∗· · ·∗µIλm∗µ(0,1]\∪λmi=1Iiwhenever I1, . . . , Imare disjoint intervals. (λl,1] To achieve (1), we start with the building blocks µλwhose entropy at small scale can be bounded below in terms of the Garsia entropy hλwhich in turn is greater than 0.44 min(Mλ, 1) by a result in [1]. Now using scale invariance and iterative convolution between these building blocks, we expect the entropy between certain scale 2−nand 2−n+1to gradually increase towards 1. The following two theorems quantify this growth. The first one is used at high entropy regime (H(µ; 2−n, 2−n+1) greater than 1 minus a small constant.) Theorem 2. Let µ and ν be compactly supported probability on R. Let α∈ (0, 1/2) and r > 0 be real numbers. Assume that for all s∈ (α3r, α−3r), 1− H(µ; s | 2s) ≤ α and 1 − H(ν; s | 2s) ≤ α. Then 1− H(µ ∗ ν; r | 2r) ≤ 108(− log α)3α2 The next theorem is used at low entropy regime (H(µ; 2−n, 2−n+1) is away from 0 but the previous theorem is not yet valid). 2882Oberwolfach Report 47/2017 Theorem 3. Given α∈ (0, 1/2), there exists c = c(α) > 0 such that the following is true. Let µ and ν be compactly supported probability on R. For any negative integers σ2< σ1< 0 and any real number β∈ (0, 1/2], if #σ ∈ Z ∩ [σ2, σ1] : 1− H(µ; 2σ| 2σ+1) < α < cβ(σ1− σ2) and H(ν; 2σ2| 2σ1) > β(σ1− σ2), then H(µ∗ ν) ≥ H(µ; 2σ2| 2σ1) +cβ(σ − log β1− σ2)− 3. References
[48] E. Breuillard and P. Varj´u, Entropy of Bernoulli convolutions and uniform exponential growth for linear groups, Preprint, arXiv:1610.09154, 2015.
[49] A.M. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102 (1962), 409-432.
[50] P. Varj´u, Absolute continuity of Bernoulli convolutions for algebraic parameters, Preprint, arXiv:1602.00261, 2016. Breuillard-Varju’s Inequality between Entropy and Mahler Measure Or Landesberg A Bernoulli convolution with parameter λP∈ (0, 1) is the distribution µλof the following random power series∞k=0±λkwhere the signs± are chosen independentlyP with equal probability. Denote the distribution of the finite sum0≤k≤n−1±λk by µ(n)λ. The random walk entropy hλof µλis defined to be: (n) H(µλ) hλ:= lim n→∞n where H(µ(n)λ) is the Shannon entropy of the finitely supported measure µ(n)λ. Let πλ(x) = ar·Πi=ri=0(x−λi) = arxr+ar−1xr−1+...+a0be the minimal polynomial in Z[x] of an algebraic number λ∈ Q, with λ1, ..., λrits Galois conjugates (including λ1= λ). The Mahler measure of λ is defined to be Mλ:=|ar| · Π|λi>1|λi|. The main theorem presented in this talk was the following due to Emmanuel Breuillard and Peter Varju [1]: Theorem 1. There exists a positive constant c > 0 such that given any algebraic number λ: c· min (1, log2λ)≤ hλ≤ min (1, log2λ) . Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2883 This constant can be taken to be c = 0.44. A special case of Hochman’s theorem on Bernoulli convolutions [3] connects the random walk entropy hλwith algebraic parameter λ∈ (12, 1) to the Hausdorff dimension of the measure µλ: h dim µλ= min1,λ log2λ−1 It is a famous conjecture by Lehmer that the Mahler measure of all algebraic numbers is uniformly bounded away from 1 whenever λ is not 0 or a root of unity. Theorem 2. If the Lehmer conjecture holds, then there exists an ε > 0 such that for every real algebraic 1− ε < λ < 1 the dimension of the Bernoulli convolution dim µλ= 1. An outline of the proof of theorem 1 was given, emphasizing the role of the Gaussian averaged entropy of a random variable. References
[51] Emmanuel Breuillard and Pter Varj´u, Entropy of Bernoulli convolutions and uniform exponential growth for linear groups, Preprint, arXiv:1510.04043v2 (2016).
[52] P´eter Varj´u, Recent progress on Bernoulli convolutions, Preprint, arXiv:1608.04210 (2016).
[53] Michael Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2), 180(2):773-822, (2014). Bernoulli convolutions with transcendental parameters, work of E. Breuillard and P. Varj´u Nicolas de Saxc´e Assume (ξn)n≥0is a sequence of±1 valued unbiased coin tosses. Given λ ∈ (0, 1),P we study the law µλof the random variablen≥0ξnλn. We present the proof of the following result. Theorem 1 (Breuillard-Varj´u). The setλ ∈ (1/2, 1) | dim µλ< 1 is contained in the closure, for the usual topology on R, of the setλ ∈ Q ∩ (1/2, 1) | dim µλ< 1, where Q denotes the algebraic closure of Q in C. 1. A question of Hochman For n≥ 1, let Pnbe the set of polynomials of degree less than n and with coefficients in−1, 0, 1, and let En=α ∈ C | ∃P ∈ Pn: P (α) = 0. In 2014, Hochman asked whether the following assertion was true: (1)∃A ≥ 0 : ∀α 6= β ∈ Pn,|α − β| ≥ e−An. As observed by Hochman, such a separation between the elements of Enhas a nice consequence on the dimension of Bernoulli convolutions. 2884Oberwolfach Report 47/2017 Theorem 2 (Hochman). If (1) holds, then for any transcendental parameter λ∈ (1/2, 1), dim µλ= 1. We now present a short proof of this theorem, based on the ideas of Breuillard and Varj´u. The following lemma will be crucial for the proof. T Lemma 1. LetP =nPn. For every ε > 0, there exists k = k(ε)≥ 0 such that every P∈ P satisfying P (0) 6= 0 has at most k roots (with multiplicity) inside the disk BC(0, 1− ε). Proof of Theorem 2, after Breuillard-Varj´u. Assume (1) holds and let λ∈ (1/2, 1) such that dim µλ< 1. By Hochman’s result on self-similar sets, this implies superexponential decay of overlaps: ∀C ≥ 0, ∃N : ∀n ≥ N, ∃Pn∈ Pn:|Pn(λ)| ≤ e−Cn. Take ε =1−λ2, let k be the number given by Lemma 1, and let C = (A+1)k+log1ε. Writing Pn= XrQn−ri=1X− αi, with|αi| > 1 − ε if i > k, we have n−rYYk e−Cn≥ |Pn(λ)| = λrε|λ − αi| ≥ λrεn−r|λ − αi|, i=1i=1 so there exists i∈ 1, . . . , k such that |λ − αi| ≤ ε−n/ke−Cn/k≤ e−n(A+1). In short, we have shown: ∀n ≥ N, ∃α ∈ En:|α − λ| ≤ e−n(A+1). To obtain a contradiction, proceed as follows: Let n0= N . Take α∈ En0such that|λ − α| := e−n0B≤ e−n0(A+1). Let n1=n0A+1B+1. Let β∈ En1be such that|λ − β| ≤ e−n1(A+1). In particular, we must have β6= α. Then |α − β| ≤ |α − λ| + |λ − β| ≤ e−n0B+ e−n1(A+1)≤ 2e−n1(A+1)< e−n1A contradicts (1), because α, β∈ En1. 2. A result of Mahler The best separation result available for the roots of polynomials inPnis due to Mahler. Theorem 3 (Mahler). If α6= β are elements in En, then|α − β| ≥ 2n−4n. Exercise 1. Using this separation result and mimicking the proof presented in the previous section, show that if the overlaps decay faster than e−Cn log nthen λ∈ Q. A slightly weaker version of the following result of Breuillard and Varj´u can be proved using Mahler’s separation bound. From now on, if I is some subset ofP R+, we let µIλdenote the law of the random variablen≥0;λn∈Iξnλn. Moreover, H(µ; r) denotes the entropy at scale r of the measure µ. Finally, for λ∈ (1/2, 1), hλdenotes the Garsia entropy of λ. Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2885 Theorem 4 (Breuillard-Varj´u). Fix λ∈ (1/2, 1). There exists c > 0 such that for all n large enough, for all r∈ (0, n−3n), the following holds. (λn,1] Assume H(µλ; r) < n. Then there exists η∈ Ensuch that|η − λ| ≤ rcand hη≤n1H(µ(λλn,1]; ) < 1. Sketch of proof. Consider the familyA = P ∈ Pn| |P (λ)| ≤ r. Using some effective version Euclidean algorithm, write the greatest common divisor D of all polynomials inA: D = P1Q1+· · · + PℓQℓ, with Pi∈ A, ℓ ≤ n, h(Qi)≤ 2n(2n)! and deg Qi≤ n. By definition of A, this shows that |D(λ)| ≤ n22n(2n)!r≤ r1/5, which implies that there exists a root η of D such that|η − λ| ≤ rc, for some small constant c > 0 depending only on λ. (The detailed argument uses Lemma 1 and is similar to the one presented in the first section.)P For a∈ N, let Ωa=(ω0, . . . , ωn−1)|ωiλi∈ [ar, (a + 1)r), so that H(µ(λλn,1]; r) =X |Ωa|log2n. 2n|Ωa| Pωi−ωi′P If ω6= ω′∈ Ωa, then P =2xjis inA, so P (η) = 0. Thereforeωiηi= P ωi′ηi, and XX |Ωa|2n 2nlog|Ω|= H(µ(λλn,1]; r). i Pm−1 Hence, hλ= infmm1H(ξiηi)≤n1H(µ(λλn,1]; r). 3. Increasing entropy using self-similarity We now want to sketch the proof of Theorem 1. The interested reader is referred to the original paper of Breuillard and Varj´u for the detailed proof. The idea is to use Varj´u’s inverse theorem for entropy and the self-similarity property of µλto increase the lower bound on entropy proved in the previous section. Let λ∈ (1/2, 1) be such that dim µλ< 1 and assume for a contradiction that there the set of η∈ Q satisfying dim µη< 1 is bounded away from λ by a small number τ . Fix ε > 0 such that dim µλ< 1− 4ε. First step. Choose n0such that: • ∀r ≤ λn0,H(µlog 1/rλ;r)≤ dim µλ+ ε(exact-dimensionality) • ∀n ≥ n0, H(µ(λλn,1]; λ10n|λn)≤ nε log 1/λ(Hochman) • n−3n00/c< τ(separation from exceptional algebraic parameters). We claim that (λn0,1] H(µλ; n−3n00|λ10n0)≥ εn0log 1/λ. 2886Oberwolfach Report 47/2017 Indeed, otherwise, we would have H(µ(λλn0,1]; n−3n00)≤ εn0log 1/λ + H(µ(λλn0,1]; λ10n0|λn0) + H(µ(λλn0,1]; λn0) ≤ n0(log 1/λ)(dim µλ+ 3ε) < n0. But by Theorem 4, there would exist η∈ Q such that |η − λ| < n−3n00/cand dim µη≤log 1/ηhη≤ dim µλ+ 4ε < 1, contradicting our absurd assumption on λ. Assuming nihas been defined, define Kiand ni+1so that one has  λKini= n−3nii ni+1= Kini This way one obtains an increasing sequence n0, n1, . . . of integers such that for each i, H(µ(λλni,1]; n−3nii|λ10ni)≥ εnilog 1/λ. Second step. Using self-similarity and Varj´u’s inverse theorem for entropy, we deduce from the above lower bound that for some c > 0 depending only on λ, for all r small enough and all p∈ 2, . . . , ni, H(µ(λλnir,λKinir]; r|λpr)≥cp. (log Ki)(log(2)Ki Third step. Since the intervals (λnir, λKinir] are disjoint, we may apply again Varj´u’s theorem and get H(µ(λλn0r,λ−KN nNr]; r|λpr)≥ cpX1. (log Ki)(log(2)Ki)2 i≤N Conclusion. To conclude, it suffices to show that the sum on the right-hand side√ diverges. For that, we check by induction that log Ki≤i + i0, where i0= (log K0)2. This is clear for i = 0, and then, write log Ki Ki) log Ki ≤ (log Ki) + C Ki C = (log Ki)(1 +) Ki ≤√i + i0+ 1. (In the last line, we used the induction hypothesis and the fact that t7→ (log t)(1+ C) is an increasing function of t for t large enough.) t Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2887 4. The refined version of the theorem In the paper by Breuillard and Varj´u, the application of Varj´u’s inverse theorem is done more carefully, in order to have a quantitative result about the approximation rate by elements of Enof the parameter λ satisfying dim µλ< 1. Their result is as follows. Theorem 5 (Breuillard-Varj´u). Let λ∈ (1/2, 1] be such that dim µλ< 1. Then, for all ε > 0, there exists A≥ 0 such that for all d0sufficiently large, there exists d∈ [d0, exp(5)(log(5)(d0) + A)] and η∈ Ed,dimµλ+εsuch that |λ − η| ≤ exp(−dlog(3)d). As a corollary, we find the first explicit examples of transcendental parameters λ for which dim µλ= 1. Corollary 1. Let λ∈ (1/2, 1) be a number such that for all n ≫ 1 and all P∈ Pn,|P (λ)| > exp(−dlog(3)d). Then dim µλ= 1. In particular, for λ∈ ln2, e−1/2, π/4, one has dim µλ= 1. Orponen’s Distance Set Theorem Demi Allen Given a planar set K⊂ R2we consider its distance set D(K) :=|x − y| : x, y ∈ K. Specifically, we are interested in how the sizes of K and D(K) are related. We begin by surveying some classical results in this area. When K is a finite set, this problem is the essence of a question asked by Erd˝os. Question (Erd˝os). What is the minimum number of different distances, f (n), determined by n distinct points in the plane? Erd˝os himself gave the following bounds for the number f (n). Theorem (Erd˝os [2], 1946). The minimum number f (n) of distances determined by n points of a plane satisfies the inequalities 3121cn n−−≤ f(n) ≤√, 42log n where c is some universal constant. A conjecture attributed to Erd˝os is that the lower bound for f (n) given above should match the upper bound in the following sense. Erd˝os Distance Conjecture. The minimum number f (n) of distances determined by n points of a plane satisfies n f (n) &√, log n 2888Oberwolfach Report 47/2017 where we write A . B to mean that there exists some universal constant c > 0 such that A < cB. Some remarkable progress has been made recently by Guth and Katz towards proving this conjecture. Theorem (Guth – Katz [5], 2015). The quantity f (n) satisfies n log n. In considerations of the “continuous” version of the question posed by Erd˝os, notions of measure and dimension make an appearance. An early result relating to the “continuous” problem is the following theorem of Steinhaus. Theorem (Steinhaus [10], 1920). If K⊂ R2is a planar set with positive 2dimensional Lebesgue measure, then the distance set D(K) contains an interval [0, ε) for some ε > 0. In the 1980s, Falconer proved a result relating the Hausdorff dimension of a set K⊂ R2with that of its distance set D(K). Theorem (Falconer [3], 1985). If K⊂ R2is any set, then dimHD(K)≥ dimHK− 1. The continuous analogue of the Erd˝os Distance Conjecture is attributed to Falconer. Falconer Distance Conjecture. If K⊂ R2is a Borel set with dimHK > 1, then D(K) has positive length, i.e.H1(D(K)) > 0 (where, for s≥ 0, Hsis the usual Hausdorff s-measure). The current best known results towards the Falconer Distance Conjecture for general sets K are due to Wolff and Bourgain. Theorem (Wolff [11], 1999). If K⊂ R2is Borel with dimHK >43, then D(K) has positive length. Theorem (Bourgain [1], 2003). If K⊂ R2is Borel with dimHK≥ 1, then dimHD(K)≥12+ ε for some (small) absolute constant ε > 0. While these theorems of Wolff and Bourgain represent the best known progress towards proving Falconer’s Distance Conjecture for general planar sets K, in recent years there has been an increased interest in proving Falconer’s Distance Conjecture or improving upon the results of Wolff and Bourgain for special classes of sets. For example, the problem for self-similar sets has been studied by Orponen [6] (2012), and self-affine sets have been considered by Ferguson, Fraser and Sahlsten [4] (2015). Very recently, the class of Ahlfors–David regular sets has also been studied by Orponen [7] (2017) and Shmerkin [8, 9] (2017). Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2889 Definition. A Borel measure µ on Rdis said to be (s, A)-Ahlfors–David regular if rs A≤ µ(B(x, r)) ≤ Ars for all x∈ spt µ and 0 < r ≤ diam(spt µ) for some constant A ≥ 1. An Hs-measurable set K⊂ Rdis said to be (s, A)-Ahlfors–David regular if 0 <Hs(K) <∞ and the restriction Hs|KofHsto K is (s, A)-Ahlfors–David regular. For this class of sets, Orponen proved the following result regarding the upper box dimension of the corresponding distance sets. Theorem (Orponen [7], 2017). Assume that∅ 6= K ⊂ R2is a boundedHsmeasurable (s, A)-Ahlfors–David regular set with s≥ 1. Then dimBD(K) = 1, where dimBdenotes the upper box dimension. We devote a large part of this talk to discussing the proof of this theorem, which relies on a careful covering argument and also employs several properties of entropy. In particular, the proof uses a projection theorem for entropy and a multi-scale decomposition of entropy to bound the entropy of projections. References
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[62] P. Shmerkin, On the Hausdorff dimension of pinned distance sets, preprint, (2017), arXiv:1706.00131
[63] H. Steinhaus, Sur les distances des points dans les ensembles de mesure positive, Fund. Math. 1 (1920), 93–104. · JFM 47.0179.02
[64] T. Wolff, Decay of circular means of Fourier transforms of measures, Internat. Math. Res. Notices (1999), no. 10, 547–567. 2890Oberwolfach Report 47/2017 Balog-Szemer´edi-Gowers Theorem Oleg Pikhurko The Balog-Szemer´edi-Gowers Theorem [2, 3] is a very powerful tool in additive combinatorics which, roughly speaking, states that any two sets A, B in an Abelian group Γ with a large fraction of sums a + b, (a, b)∈ A × B, concentrated on few values necessarily have large subsets A′⊆ A and B′⊆ B with A′+ B′:=a + b | a ∈ A′, b∈ B′, the set of all possible sums, having small size. It was first proved by Balog and Szemer´edi [2] using Szemer´edi’s Regularity Lemma for graphs, with the quantitative dependences between the parameters being rather bad. Later, Gowers [3] found a proof that avoided regularity and gave polynomial dependence between the parameters. A detailed discussion of the Balog-Szemer´edi-Gowers Theorem can be found in the book of Tao and Vu [4]. One version of the Balog-Szemer´edi-Gowers Theorem is as follows (see [4, Theorem 2.29]). Given G⊆ A × B, define the partial sumset G A+ B :=a + b | (a, b) ∈ G. Theorem 1 (Balog-Szemer´edi-Gowers Theorem (Symmetric Version)). Let A, B be subsets of an Abelian group Γ. Let G⊆ A × B and reals K ≥ 1 and K′> 0 satisfy: |A| · |B|, |G| ≥ K G A+ B ≤ K′|A|1/2|B|1/2. Then there are A′⊆ A and B′⊆ B such that |A′| ≥√|A|, 42 K |B′| ≥|B|, 4K |A′+ B′| ≤ 212K5(K′)3|A|1/2|B|1/2. In fact, there are other properties that can be attained in the conclusion of Theorem 1. These are discussed by Balog [1] who concentrates on the case when |A|, |B|, A+ BG ≤ N and |G| ≥ N2/K, and shows that one can attain|A′−B′| ≤ O(K7N ),|A′− A′| ≤ O(K5N ),|A′+ A′| ≤ O(K7N ), and|(A′× B′)∩ G| ≥ Ω(N2/K4). The above theorem is particularly useful if the sizes of A and B are within constant factor of each other, since then it can be combined with the FreimanRuzsa Theorem to derive a very strong structural information about the obtained Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2891 sets A′and B′. If|A| is significantly larger than |B|, then the direct application of Theorem 1 to the sets A and B is not very useful as then KK′≥ A+ B|A|1/2|B|1/2≥|A|1/2 |G||B|1/2 has to be large. There is an “asymmetric” version of the Balog-Szemer´edi-Gowers Theorem designed for such cases, see Theorem 2.35 in [4], which can be derived with some work from Theorem 1. For its statement and proof, see Section 2.6 in [4]. References
[65] A. Balog, Many additive quadruples, Additive combinatorics, CRM Proc. Lecture Notes, vol. 43, Amer. Math. Soc., Providence, RI, 2007, pp. 39–49. · Zbl 1181.11070
[66] A. Balog and E. Szemer´edi, A statistical theorem on set addition, Combinatorica 14 (1994), 263–268. · Zbl 0812.11017
[67] W. T. Gowers, A new proof of Szemer´edi’s theorem for arithmetic progressions of length four, Geom. Func. Analysis 8 (1998), 529–551. · Zbl 0907.11005
[68] T. Tao and V. Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge Univ. Press, 2006. Bourgain’s sum-product and projection theorems. Part I Giorgis Petridis The topic of this expository talk is a sum-product result of Bourgain, where the “largeness” of a finite set is measured by covering number. Applications of Borgain’s theorem are discussed in the subsequent talk. The classical sum-product phenomenon relies on “largeness” to measure the size of a set. The best known of its many manifestations is that every finite set must either have a large number of pairwise sums or a large number of pairwise products. Let A⊂ R and denote by A + A =a1+ a2: a1, a2∈ A its sum set and by AA the similarly defined product set. Konyagin and Shkredov [6], building on work of Solymosi [7], prove that at least one of|A + A| and |AA| must be much larger than |A|: there exist positive constants c and ε, with ε approximately equal to 1/5, 000, such that for all finite sets A⊂ R we have max|A + A|, |AA| ≥ c|A|4/3+ε. The conjectured exponent is 2. A different manifestation is to prove that sets like AA + AA =a1a2+ a3a4: a1, . . . , a4∈ A have cardinality much larger than|A|. The best known result is due to Iosevich, Roche-Newton, and Rudnev, who show (up to logarithms) that|AA + AA| ≥ c|A|19/12
[69] . 2892Oberwolfach Report 47/2017 Bourgain, inspired by questions in geometric measure theory, considered a sumproduct question for finite sets A⊂ δZ = δn : n ∈ Z, for some absolute δ > 0. He uses the δ-covering number to measure how “large” sets like A, A + A, AA, . . . are. Given a finite set S⊆ R, he denotes by N(A, δ) the minimum number of intervals in δZ that S intersects. While it is true that for A⊂ δZ, we have N(A, δ) = |A|, and even N(A + A, δ) = |A + A|, it is not true that N (AA, δ) =|AA|. We do however have |AA| > N(AA, δ). In this sense Bourgain proved a strong sum-product theorem by establishing something like: maxN(A + A, δ), N(AA, δ) ≥ cN(A, δ)1+ε. Before making the above heuristic statement precise, we note that taking A = δZ∩ [1, x) (for any x > δ) shows that the above is false, because all three covering numbers are of the order of magnitude of xδ−1. This means that some care must be taken to make sure that the sets considered are not similar to the intersection of δZ and an interval. Here is Bourgain’s theorem from [2]. ℓ(·) denotes Lebesgue measure. Theorem 1 (Bourgain). Let α > 0 and κ be positive absolute constants. There exist ε0and ε1with the following property. For all δ > 0 and all A⊂ δZ of cardinality |A| ≥ δαsuch that |A ∩ I| ≤ ℓ(I)κ|A| for all intervals I of length δ < ℓ(I) < δε0, we have maxN(A + A, δ), N(AA, δ) ≥ δ−ε1N (A, δ). The theorem is deduced from the following more general result from [2], which Bourgain also applies to projection theorems for fractal sets. Theorem 2 (Bourgain). Let α > 0 and κ be positive absolute constants. There exist ε0and ε1with the following property. For all δ > 0, and all probability measures µ supported on [0, 2] with the property µ(I)≤ ℓ(I)κ for all intervals I of length δ < ℓ(I) < δε0, and all A⊂ δZ of cardinality |A| ≥ δα such that |A ∩ I| ≤ ℓ(I)κ|A| for all intervals I of length δ < ℓ(I) < δε0, there exists x∈ support(µ) such that N (A + xA, δ)≥ δ−ε1N (A, δ). The deduction of Theorem 1 from Theorem 2 relies on methods from additive combinatorics presented in the seventh talk of this study group and is reminiscent of arguments of Gowers in [3] and Katz and Tao in [5]. A key step is to show that if both N (A + A, δ) and N (AA, δ) are comparable to N (A, δ), then N (AA + AA, δ) is also comparable to N (A, δ). This leads to a contradiction by a robust generalisation of the result of Iosevich, Roche-Newton, and Rudnev [4]. Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2893 The proof of Theorem 2 is very intricate. It also uses techniques from additive combinatorics, but the key ingredient is a tree-structure theorem, which in a way is an inverse result for an example discussed in the eight talk of this workshop. Bourgain proves that if N (A + A, δ) is comparable to N (A, δ), then A contains a fairly large subset with a tree-structure similar to that of the sets discussed in the eighth lecture on Hochman’s inverse theorem for entropy. This inverse result was used by Shmerkin in the work discussed in the final two talks of this workshop. References
[70] J. Bourgain, On the Erd˝os-Volkmann and Katz-Tao ring conjectures, Geom. Funct. Anal. 13(2003), 334–365.
[71] J. Bourgain, The discretized sum-product and projection theorems, J. Anal. Math. 112 (2010), 193–1236. · Zbl 1234.11012
[72] W. T. Gowers, A new proof of Szemer´edi’s theorem for arithmetic progressions of length four., Geom. Funct. Anal. 8 (1998), 529–551. · Zbl 0907.11005
[73] A. Iosevich, O. Roche-Newton, M Rudnev, On discrete values of bilinear forms, preprint. · Zbl 1406.52034
[74] N. Katz, T. Tao, Some connections between the Falconer and Furstenburg conjectures, New York J. Math. 7 (2001), 148–187.
[75] S. V. Konyagin, I. D. Shkredov, New results on sum-products in R, Tr. Mat. Inst. Steklova 294(2016), 87–98.
[76] J. Solymosi, Bounding multiplicative energy by the sumset, Adv. Math. 222 (2009), 402– 4088. Bourgain’s sum-product and projection theorems. Part II. Andras Mathe Given a Borel set in the plane of a certain (Hausdorff) dimension, what can we say about the (Hausdorff) dimension of its orthogonal projections to various lines in the plane? Answers to questions of this type are called projection theorems. This talk presented Bourgain’s projection theorem and its background, and indicated how the proof follows from his sum-product theorems [1]. The sumproduct theorems were the subject of the previous talk. Let πθdenote the orthogonal projection from the plane to the line in direction θ. Hausdorff dimension is denoted by dimH. The classical projection theorem is the following. Theorem 1 (Marstrand). Let E⊂ R2be a Borel set with dimHE = α. Then for almost every direction θ, dimHπθ(E) = min(1, α). Moreover, if α > 1 then πθ(E) has positive Lebesgue measure for almost all θ. Theorem 2 (Falconer). Let E⊂ R2be Borel with dimHE = 1 + t. Then the dimension of the “exceptional set of directions” is dimHθ : dimHπθ(E) < 1 ≤ 1 − t. 2894Oberwolfach Report 47/2017 Upper box dimension of product sets satisfies the inequality dimB(X× Y ) ≤ dimBX + dimBY. From this we immediately obtain Observation 3. Let E⊂ R2have upper box dimension α. Then there is at most one direction in which the projection has upper box dimension less than α/2. One could also replace box dimension with Hausdorff dimension in the previous observation: then the exceptional set of directions will be of zero Hausdorff dimension. In the other direction, we have the following construction. Theorem 4 (Kaufman, Mattila). Let 0≤ γ ≤ 1, γ ≤ α ≤ 2 − γ. There is a Borel set E⊂ R2such that dimHE = α and dimHθ : dimHπθ(E) < (α + γ)/2 = γ. It is not known whether this result is sharp in general. It is sharp if γ = 0 or if α = 1 + t, γ = 1− t, or if α = γ. Now we state Bourgain’s projection theorem. Theorem 5 (Bourgain [1]). For every 0 < α < 2 and γ > 0 there is ε > 0 with the following property. Let E⊂ R2be Borel with dimHE≥ α. Then dimHθ : dimHπθ(E) < α/2 + ε ≤ γ. The proof of this theorem relies on the following discretised sum-product theorem. Theorem 6 (Bourgain [1]). Given 0 < σ < 1 and γ > 0, there exist ε > 0 and ε0> 0 such that the following holds for δ > 0 sufficiently small. Let µ be a probability measure on [0, 1] satisfying µ([x, y])≤ |y − x|γ whenever δ <|y−x| < δε0. Let A⊂ [1, 2] be a discrete set consisting of δ-separated points satisfying |A| ≥ δ−σ such that also |A ∩ [x, y]| ≤ |y − x|γ|A| whenever δ <|y − x| < δε0. Then there exists x∈ suppµ such that N (A + xA, δ) > δ−ε|A| where N (X, δ) denotes the minimum number of δ-intervals needed to cover the set X. In the non-discrete world this theorem roughly says the following: For every 0 < σ < 1 and γ > 0 there is ε > 0 such that whenever A⊂ R has dimension at least σ, and S⊂ R has dimension at least γ, then there is x ∈ S such that A + xA has dimension at least σ + ε. However, this is vague (we did not specify what we Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2895 mean by dimension), and the discrete version is actually much more powerful than this analogue suggests. Notice that A + xA is essentially the projection of A× A to the line of slope x. Theorem 7 (Bourgain [1]). Given 0 < α < 2, α′> 0 and γ > 0, there exist ε0> 0 and ε > 0 such that the following holds. Let µ be a probability measure on S1(set of directions in the plane) such that the µ-measure of every interval (arc) of length ℓ is at most Cℓγ. [This implies that the support of µ has Hausdorff dimension at least γ.] Let δ > 0 be chosen sufficiently small and let E⊂ [1, 2] × [1, 2] be a δ-separated set satisfying |E| = δ−α and |E ∩ B(x, r)| ≤ rα′|E| for every disc B(x, r) of radius r with δ < r < δε0. Then there exists θ∈ suppµ such that N (πθ(E), δ)≥ δ−(α/2+ε). Theorem 7 is proved from Theorem 6 using the Balog–Szemer´edi–Gowers theorem (and sumset inequalities). This proof may be described (slightly incorrectly) in the following way. Assume there is no such θ. Then take two directions in the support of µ; these guarantee two small projections. That is, E (after applying an affine map and modulo moving points by distances at most δ) is contained in a product B× B where B is a δ-separated set and |E| ≈ |B|2. In this setting, the Balog–Szemer´edi–Gowers theorem can be effectively applied to E⊂ B × B to yield a product set A× A. It can be shown that A × A can be covered by a few translates of E and since E has small projections, so does A× A. However, this means that A contradicts Theorem 6. Theorem 5 (Bourgain’s projection theorem) follows from Theorem 7 using standard techniques and some ideas that are perhaps less standard and not explained in [1]. See [3] for details. References
[77] J. Bourgain. The discretized sum-product and projection theorems. J. Anal. Math. 112 (2010), 193–236. · Zbl 1234.11012
[78] K.J. Falconer. Hausdorff dimension and the exceptional set of projections. Mathematika 29 (1982), no. 1, 109–115. · Zbl 0477.28004
[79] W. He. Sums, products and projections of discretized sets. PhD thesis (2017), Universit´e Paris-Sud, Orsay.
[80] R. Kaufman, P. Mattila. Hausdorff dimension and exceptional sets of linear transformations, Ann. Acad. Sci. Fenn. A Math. 1 (1975), 387–392. · Zbl 0348.28020
[81] J.M. Marstrand. Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. (3) 4 (1954), 257–302. 2896Oberwolfach Report 47/2017 Shmerkin’s theorem on Lpdimensions, and applications. Part I Alex Iosevich The purpose of this talk is to present the first half of Pablo Shmerkin’s paper, entitled ”On the Furstenberg intersection conjecture, self-similar measures and the Lqnorms of convolutions”. This paper combines analytic, geometric and combinatorial methods to prove several open conjectures involving intersection of sets, Bernoulli convolutions and others. More precisely, the author studies measures possessing a self-similar structure, which he calls dynamically driven selfsimilar measures, and contain some classical self-similar measures such as Bernoulli convolutions as special cases. The main result of the paper gives an expression for the Lqdimensions of such dynamically driven self-similar measures. As an application, the the celebrated Furstenberg intersection conjecture established. It says that If A,B are closed subsets of the circle [0, 1), invariant under Tp,Tq respectively, with p and q multiplicatively independent, then dimH(A∩ B) ≤ maxdimH(A) + dimH(B)− 1, 0, where Tp(x) = px mod 1. In this talk we shall present the notion of the Lq-dimensional and explain how it be used to study the Furstenberg intersection conjecture and related problems about the Bernoulli convolutions. The key is to relate the notion of the Lqdimension to the Frostman exponents governing the size of small balls and the sizes of fibers. Let µ be a Borel probability measure on [0, 1]. Consider the family of intervals of length 2−mj2−m, (j + 1)2−m, j ∈ Z, denoted by Dm. The Lq dimension of the measure µ is defined to be P logI∈Dµq(I) lim inf−m. m→∞m(q− 1) Once can check that, roughly speaking, that under the assumption that under the assumption that the Lqdimension is s, it follows that µ(B(x, r))≤ Cr(1−1/q)s, thus linking the notion of the Lq-dimension and the usual Hausdorff dimension. One can also check that under the same assumption, the upper box dimension of the fibers under Lipschitz maps cannot exceed s− α, where α is the Frostman exponent of µ under the said Lipschitz map. These elegant observation lie at the core of the intricate web spun by the author which allows him to knock off the intersection conjecture mentioned above along with several related results. We shall describe the set of dynamically self-driven models and their Lq-dimension, building up to the main result of the Shmerkin paper (Theorem 11.1) from which everything else is derived. These measures are modeled after p-Cantor sets, that is sets whose base p expansion digits lie in a given subset of0, 1, . . . , p − 1, p prime. In the process we shall explain how the basic ideas described above are combined with the notion of dynamically self-driven measures to set up the main mechanism of the paper. Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2897 References
[82] P. Shmerkin, On Furstenberg’s intersection conjecture, self-similar measures, and the Lq norms of convolutions, Preprint 2016. Lq-spectrum of homogeneous self-similar measures and inverse theorem for the decay of Lq-norms under convolutions. After P. Shmerkin Julien Barral Let λ∈ (0, 1), b ∈ N≥2,A = 0, . . . , b − 1, p = (pi)i∈Aa probability vector, and (ti)i∈A∈ Rb. For a∈ R∗+, define Sa: x∈ R 7→ ax. For i ∈ A define ϕi: x∈ R 7→ Sλ(x)+ti. For n∈ N and I = i1· · · in∈ Andefine ϕI= ϕi1◦· · ·◦ϕin and pI= pi1· · · pin. Also define  XX (1)µn=∗n−1i=0Sλi∗pjδtj=pIδϕI(0). j∈AI∈An For m∈ N, define Dm=[k2−m, (k + 1)2−m) : k∈ Z. Denote by µ the unique homogeneous self-similar probability measure associated with p and the IFSϕi: i∈ A, i.e. the unique Borel probability measure on R such thatX µ =piµ◦ ϕ−1i. i∈A Recall that µ is supported by the unique non-empty compact set KS⊂ R such that K =i∈Aϕi(K). Without loss of generality we assume that K⊂ [0, 1]. The measure µ can also be written in the following form:  X (2)µ =∗∞i=0Sλi∗pjδtj= µn∗ (Sλn∗µ). j∈A Define the Lq-spectrum of µ as the concave mapping 1X (3)τµ: q∈ R+7→ lim−log2µ(I)q m→∞m I∈Dm (that the limit does exist was proved in [4] and follows from a submultiplicative property). The convolution structure (2) can be exploited to prove the following deep theorem. Theorem 1 (P. Shmerkin [5]). Either ∆n= min|ϕI(0)− ϕJ(0)| : I 6= J ∈ An converges super-exponentially to 0 as n→ ∞, or P! logj∈Apqj ∀ q ≥ 1, τµ(q) = τ (q) := minq− 1,. log λ Remark 2. (1) Theorem 1 is a special case of the main result of [5] on dynamically driven self-similar measures. (2) Under the open set condition, Theorem 1 is known and it holds in any dimension [3]. 2898Oberwolfach Report 47/2017 (3) The measure µ is exact dimensional [1], hence its entropy dimension H(µ) is well defined and equals dim(µ). On the other hand, it is easily seen that H(µ)≤ τ′(1+) always hold, and it is always true that dim(µ)≥ τµ′(1+). Consequently, Theorem 1 implies M. Hochman’s result on the dimension of homogeneous selfsimilar measures [2]. For n∈ N, let πn: x∈ AN7→ ϕx1···xn(0); then define π = limn→∞πn. Denoting by ρ the Bernoulli product measurePi∈Apiδi⊗N, we have µ = π∗ρ, µn= πn∗ρ, andkπ − πnk∞= O(λn), from which it follows that for all q > 1, there exists Cq> 0 such that XXX (4)Cq−1µ(I)q≤µn(I)q≤ Cqµ(I)q,∀ n ∈ N, I∈Dm(n)I∈Dm(n)I∈Dm(n) where 2−m(n)≤ λ−n< 2−m(n)+1. Thus, one can focus on Sn(q) =PI∈Dµn(I)q m(n) to get Theorem 1. In particular, since it is easy to see using the subadditivity ofP x7→ xqthat−m(n)1log2Sn(q)≤loglog λj∈Apqjfor all q > 1, while on the other hand the upper bound q− 1 holds for any probability measure, we get τµ(q)≤ τ(q). When ∆ndoes not converge super-exponentially to 0, the opposite inequality follows from the following remarkable fact. Theorem 3 ([5]). Let q > 1. Suppose that τPµ(q) < q− 1 and τµ′(q) exists. Then, for all R∈ N, limn→∞I∈Dµn(I)q= τµ(q). Rm(n) Suppose that ∆ndoes not converge super-exponentially to 0. Then, thereP exists R∈ N such that for infinitely many n, for all q > 1,I∈Dµn(I)q= J∈AnpqJ= Pi∈Apqin. Consequently, due to Theorem 3 and (4), for any q of the dense subset of (1,∞) over which τµis differentiable, the equality τµ(q) = τ (q) holds, and it extends to [1,∞) by continuity. Theorem 3 is a consequence of (2), (4), and a flattening theorem for Lq-norms of discretized version of µ. Before stating this result we need some new definitions. If m∈ N, a 2−m-measure is a probability measure supported on 2−mZ ∩ [0, 1].For any compactly supported Radon measure ν on R, define ν(m)= P k∈Zν([k2−m, (k + 1)2−m)δk2−m. If qP≥ 1, the Lqnorm of any finitely supported measure ρ is defined bykρkqq=y∈supp(ρ)ρ(y)q. The Young inequality kρ ∗ ν(m)kq≤ kρk1kν(m)kqholds. Remind that due to (3):∀ ǫ′> 0, for m large enough,kµ(m)kqq≥ 2−(τµ(q)+ǫ′)m. Theorem 4 (Flattening property for Lq-norms of µ(m)under convolution [5]). Let σ > 0 and q > 1 such that τ′(q) exists and τ (q) < q− 1. Then, there exists ǫ = ǫ(σ, q) > 0 such that for m large enough, if ν is a 2−m-measure and kνkqq≤ 2−σ(q−1)m, thenkν ∗ µ(m)kqq≤ 2−(τµ(q)+ǫ)m≤ 2−ǫm/2kµ(m)kqq. Theorem 4 is proved by contradiction. The proof combines fine large deviations estimates associated with µ at “temperature” 1/q when q > 1, τµ′(q) exists, and Arbeitsgemeinschaft: Combinatorics, Entropy, and Fractal Geometry2899 qτµ′(q)− τµ(q) < 1 (which holds when τµ(q) < q− 1), and an inverse theorem for the decay of Lq-norms under convolutions (Theorem 5) that we state after introducing new definitions. If A⊂ R and s ∈ N, define Ds(A) =I ∈ Ds: I∩A 6= ∅ and Ns(A) =|Ds(A)|. If x∈ R, Ds(x) stands for the unique I∈ Dssuch that x∈ I. If a > 0 and I is an interval, aI stands for the interval with the same center as I and length a|I|. If m∈ N, a 2−m-set is a subset of 2−mZ ∩ [0, 1]. Let D, ℓ∈ N and set m = Dℓ. Given R = (Rs)0≤s≤ℓ−1∈ [1, 2D]ℓ, say that a 2−m-set A is (D, ℓ, R)-uniform ifN(s+1)D(A∩ I) = Rsfor all 0≤ s ≤ ℓ − 1 and I∈ DsD(A), i.e. A has the structure of a spherical tree of height ℓ, with branching number Rsat generation sD, 0≤ s ≤ ℓ − 1. The following result gives a very precise structural description of two 2−mmeasures ρ and ν wheneverkρ ∗ νkqq≥ 2−mǫqkρkqq. In particular, it tells that along an arithmetic sequence of scales, either a large proportion of ν looks atomic, or ρ is distributed rather uniformly when restricted to a subset which carries a large proportion of the Lqnorm of ρ. This has a flavour similar to that of the inverse theorem for the growth of entropy established in [2]. Theorem 5 ([5]). Let q > 1, δ > 0 and D0∈ N. There are ǫ > 0 and D ≥ D0, such that if ℓ is large enough, m = ℓD and ρ and ν are 2−m-measures such that kρ ∗ νkq≥ 2−mǫkρkq, the following holds: after translating ρ and ν by appropriate numbers of the form k2−m, there exist A⊂ supp(ρ) and B ⊂ supp(ν) such that (i)kρ|Akq≥ 2−mδkρkqand ν(B)≥ 2−mδ; (ii) ρ(y) ≤ 2ρ(x) for all x, y ∈ A and ν(y) ≤ 2ν(x) for all x, y ∈ B; (iii) x∈12DsD(x) for all x∈ A ∪ B and 0 ≤ s ≤ ℓ − 1; (iv) There exists RAand RBsuch that A and B are respectively (D, ℓ, RA) and (D, ℓ, RB)-uniform. (v) For all 0≤ s ≤ ℓ − 1, either RBs= 1 or RAs≥ 2(1−δ). log(kνkqq)log(kρkq q− 1−δm ≤ D·|0 ≤ s ≤ ℓ−1 : RsA≥ 2(1−δ)| ≤ −q− 1q)+δm. To prove Theorem 5, P. Shmerkin first exploits the inequalitykρ ∗ νkq≥ 2−mǫkρkqto construct sets A1and B1such that (i) and (ii) hold, and the additive energy of A1and A2fulfills the assumption of a slightly simplified version of the asymmetric Balog-Szemeredi-Gowers theorem. He also establishes a beautiful refinement of Bourgain’s structural result for small doubling sets. This result is then applied to the small doubling set produced by asymmetric B-S-G theorem to get the desired sets A and B, after a series of delicate manipulations. References
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