zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Chowla’s cosine problem. (English) Zbl 1209.43003

Let G be an abelian group, to be thought of as discrete. For a finite symmetric subset AG, one can ask how large the negative Fourier coefficients of the indicator function 1 A can be. (Note that the largest positive Fourier coefficient is trivially equal to the size of the set A. Also, since the set A is symmetric, the Fourier coefficients of 1 A are real, thus the question regarding the maximal negative value of the Fourier transform is well defined.)

We shall give a very brief history of the problem before stating the results of this paper. For a set AG as above, define

M G (A)=sup γG ^ -1 A ^(γ)·

S. Chowla [J. Reine Angew. Math. 217, 128–132 (1965; Zbl 0127.02104)] asked for a lower bound on M (A). A simple averaging argument and the Littlewood conjecture [S. V. Konyagin, Izv. Akad. Nauk SSSR, Ser. Mat. 45, 243–265 (1981; Zbl 0493.42004); O. C. McGehee, L. Pigno and B. Smith, Ann. Math. (2) 113, 613–618 (1981; Zbl 0473.42001)] imply that M (A)=Ω(log|A|). The best known bound is due to I. Z. Ruzsa [Acta Arith. 111, No. 2, 179–186 (2004; Zbl 1154.11312)] and of the form M (A)=exp(Ω(log|A|)).

Littlewood’s conjecture has recently been extended to abelian groups other than by B. Green and S. Konyagin [Can. J. Math. 61, No. 1, 141-164 (2009; Zbl 1232.11013)]. Their results imply, for example, that M /p (A)=log Ω(1) |A| for p a prime, provided that |A|=(p+1)/2.

In the current paper the author is able to improve on this and obtain the bound M /p (A)=Ω(p 1/3 ), again provided that |A|=(p+1)/2. For comparison, J. Spencer showed in [Trans. Am. Math. Soc. 289, 679–706 (1985; Zbl 0577.05018)] that there exist sets A/p of size (p+1)/2 such that M /p (A)=O(p 1/2 ).

In more general abelian groups there is a simple but devastating obstacle to the obvious extension of the above result: if H is a finite subgroup of G, then M G (H)=0. The author therefore proves the following refinement, which is easily seen to imply the statement for /p above.

Theorem. Suppose that G is a finite abelian group and A a symmetric subset of G with |A|=Ω(|G|). Then there is a subgroup HG such that

M G (A)=|AΔH| Ω(1) ·

The example of a set A consisting of a large finite subgroup together with a handful of other points shows that this result is best possible up to a power.

Finally, in order to remove the hypothesis on the density of A in the theorem above, the author allows unions of subgroups to enter the picture, but we shall not state the full result here.

The paper, and in particular the introduction, is beautifully written. It draws on a number of techniques from [B. Green and T. Sanders, Ann. Math. (2) 168, No. 3, 1025–1054 (2008; Zbl 1170.43003)], including approximately 0,1-valued functions and so-called Bourgain systems, and employs an iterative method of proof.

43A25Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
11K38Irregularities of distribution
11K06General theory of distribution modulo 1
[1]A. Balog and E. Szemerédi, A statistical theorem of set addition, Combinatorica 14 (1994), 263–268. · Zbl 0812.11017 · doi:10.1007/BF01212974
[2]J. P. Bell, P. B. Borwein and L. B. Richmond, Growth of the product j=1 n (1 a j), Acta Arithmetica 86 (1998), 155–170.
[3]J. Bourgain, Sur le minimum d’une somme de cosinus, Acta Arithmetica 45 (1986), 381–389.
[4]J. Bourgain, On the distributions of the Fourier spectrum of Boolean functions, Israel Journal of Mathematics 131 (2002), 269–276. · Zbl 1021.43004 · doi:10.1007/BF02785861
[5]S. Chowla, Some applications of a method of A. Selberg, Journal für die Reine und Angewandte Mathematik 217 (1965), 128–132. · Zbl 0127.02104 · doi:10.1515/crll.1965.217.128
[6]P. Erdos and G. Szekeres, On the product k=1 n (1 a k), Acad. Serbe Sci. Publ. Inst. Math. 13 (1959), 29–34.
[7]G. A. Freĭman, Foundations of a Structural Theory of Set Addition, American Mathematical Society, Providence, RI, 1973, Translated from the Russian, Translations of Mathematical Monographs, Vol. 37.
[8]W. T. Gowers, A new proof of Szemerédi’s theorem for arithmetic progressions of length four, Geometric and Functional Analysis 8 (1998), 529–551. · Zbl 0907.11005 · doi:10.1007/s000390050065
[9]B. J. Green and S. V. Konyagin, On the Littlewood problem modulo a prime, Canadian Journal of Mathematics 61 (2009), 141–164. · Zbl 1232.11013 · doi:10.4153/CJM-2009-007-4
[10]B. J. Green and I. Z. Ruzsa, Freĭman’s theorem in an arbitrary abelian group, Journal of the London Mathematical Society. Second Series 75 (2007), 163–175. · Zbl 1133.11058 · doi:10.1112/jlms/jdl021
[11]B. J. Green and T. Sanders, A quantitative version of the idempotent theorem in harmonic analysis, Annals of Mathematics (2) 168 (2008), 1025–1054. · Zbl 1170.43003 · doi:10.4007/annals.2008.168.1025
[12]B. J. Green and T. C. Tao, A note on the Freĭman and Balog-Szemerédi-Gowers theorems in finite fields, Journal of the Australian Mathematical Society 86 (2009), 61–74. · Zbl 1232.11014 · doi:10.1017/S1446788708000359
[13]S. V. Konyagin, On the Littlewood problem, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 45 (1981), 243–265, 463.
[14]S. V. Konyagin and V. F. Lev, On the distribution of exponential sums, Integers. Electronic Journal of Combinatorial Number Theory (2000), A1, 11 pp. (electronic).
[15]S. V. Konyagin and V. F. Lev, Character sums in complex half-planes, Journal de Théorie des Nombres de Bordeaux 16 (2004), 587–606.
[16]O. C. McGehee, L. Pigno and B. Smith, Hardy’s inequality and the L 1 norm of exponential sums, Annals of Mathematics. Second Series 113 (1981), 613–618. · Zbl 0473.42001 · doi:10.2307/2007000
[17]K. F. Roth, On cosine polynomials corresponding to sets of integers, Acta Arithmetica 24 (1973), 87–98, Collection of articles dedicated to Carl Ludwig Siegel on the occasion of his seventy-fifth birthday, I.
[18]W. Rudin, Fourier analysis on groups, Wiley Classics Library, John Wiley & Sons Inc., New York, 1990, Reprint of the 1962 original, A Wiley-Interscience Publication.
[19]I. Z. Ruzsa, Negative values of cosine sums, Acta Arithmetica 111 (2004), 179–186. · Zbl 1154.11312 · doi:10.4064/aa111-2-6
[20]J. Spencer, Six standard deviations suffice, Transactions of the American Mathematical Society 289 (1985), 679–706. · doi:10.1090/S0002-9947-1985-0784009-0
[21]T. Tao and V. Vu, Additive Combinatorics, Vol. 105 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2006.