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Wiener’s ‘closure of translates’ problem and Piatetski-Shapiro’s uniqueness phenomenon. (English) Zbl 1231.42003

For \(1 \leq p < \infty\), a sequence \({\mathbf c} = (c_n) \in \ell_p ({\mathbb Z})\) is said to be cyclic in \(\ell_p ({\mathbb Z})\) if the linear span of the translates of \({\mathbf c}\) is dense in \(\ell_p ({\mathbb Z})\). N. Wiener [“Tauberian theorems”, Ann. Math. (2) 33, 1–100 (1932; Zbl 0004.05905)] proved, as a tool towards his Tauberian theorems, that: (1) a sequence \({\mathbf c} = (c_n)\) is cyclic in \(\ell_2 ({\mathbb Z})\) if and only if its Fourier transform \(\widehat {\mathbf c} (t) = \sum_{n \in {\mathbb Z}} c_n \text{e}^{int}\) is nonzero almost everywhere; (2) \({\mathbf c}\) is cyclic in \(\ell_1 ({\mathbb Z})\) if and only if \(\widehat {\mathbf c}\) is nonzero everywhere (though he actually stated these results for \(L^2 ({\mathbb R})\) and \(L^1 ({\mathbb R})\)), and he conjectured that the cyclicity of vectors \({\mathbf c}\) in \(\ell_p ({\mathbb Z})\) could be characterized by the smallness of the set \(Z_{\widehat {\mathbf c}} = \{ t \in {\mathbb T}\,;\;\widehat {\mathbf c} (t) = 0\}\) of the zeros of the Fourier transform \(\widehat {\mathbf c}\) of \({\mathbf c}\), at least for \(1 \leq p \leq 2\) (in their book, [J.-P. Kahane and R. Salem, Ensembles parfaits et séries trigonométriques, Actualités Scientifiques et Industrielles 1301, Paris, Hermann & Cie (1963; Zbl 0112.29304)] referred to this question as posed by A. Beurling in 1947). In the paper under review, the authors disprove this conjecture: they show that, for every \(1 < p < 2\), there exist \({\mathbf c}_1, {\mathbf c}_2 \in \ell_1 ({\mathbb Z})\) such that \(Z_{\widehat {\mathbf c}_1} = Z_{\widehat {\mathbf c}_2}\), but \({\mathbf c}_1\) is cyclic in \(\ell_p ({\mathbb Z})\), even though \({\mathbf c}_2\) is not. A slightly weaker version was announced by the authors previously [“No characterization of generators in \(\ell ^p\) \((1 < p < 2)\) by zero set of Fourier transform”, C. R. Math. Acad. Sci. Paris 346, No. 11–12, 645–648 (2008; Zbl 1155.46015)]. Actually, this result is a straightforward consequence of the following theorem (Theorem 1): for every \(1 < p < 2\), there exists a compact set \(K \subseteq {\mathbb T}\) such that: (i) if \(\sum_{n \in {\mathbb Z}} |n|^\varepsilon |c_n| < \infty\) for some \(\varepsilon > 0\) and \(\widehat {\mathbf c} = 0\) on \(K\), then \({\mathbf c}\) is not cyclic in \(\ell_p ({\mathbb Z})\); (ii) there exists \({\mathbf c}_1 \in \ell_1 ({\mathbb Z})\) such that \(\widehat {\mathbf c}_1 = 0\) on \(K\) and \({\mathbf c}_1\) is cyclic in \(\ell_p ({\mathbb Z})\).
To prove this theorem, the authors look at the “other side” of the Fourier transform. Recall that every bounded sequence \({\mathbf c} \in \ell_\infty ({\mathbb Z})\) defines a distribution \(S\) on \({\mathbb T}\) such that \(\widehat S = {\mathbf c}\), called a pseudo-measure. This pseudo-measure is called a pseudo-function if \(\widehat S \in c_0 ({\mathbb Z})\). For \(1 \leq q < \infty\), let \(A_q ({\mathbb T})\) be the space of pseudo-functions \(S\) such that \(\widehat S \in \ell_q ({\mathbb Z})\). As a consequence of their Theorem 1, they get that for every \(q > 2\), there exists a compact set \(K \subseteq {\mathbb T}\) such that: (i’) there exists a nonzero \(S \in A_q ({\mathbb T})\) such that \(\text{supp}\,S \subseteq K\); (ii’) there is no nonzero measure \(\mu \in A_q ({\mathbb T})\) such that \(\text{supp}\,\mu \subseteq K\) (this result was announced by the authors in [“Piatetski-Shapiro phenomenon in the uniqueness problem”, C. R. Math. Acad. Sci. Paris 340, No. 11, 793–798 (2005; Zbl 1063.43006)]). Actually, (i’) is equivalent to (i). This corollary gives a link to the classical notion of sets of uniqueness and sets of multiplicity. Indeed, it is known that a compact set \(K \subseteq {\mathbb T}\) is a set of uniqueness if and only if no nonzero pseudo-function is supported by \(K\). Hence the corollary says that a counterexample “must” be a set of multiplicity of a particular type which were constructed for the first time by I. Piatetski-Shapiro in 1954: compact sets of multiplicity which support no nonzero Rajchman measure \(\mu\) (i.e., \(\widehat \mu \in c_0 ({\mathbb Z})\)). Later, in 1973, T. Körner and R. Kaufman, independently, constructed such sets which are moreover Helson sets. It appears that such sets are the good ones: the authors show that for every (compact) Helson set \(K\), there is a function \(g \in A ({\mathbb T})\) (the Wiener algbra) vanishing on \(K\) and cyclic in \(A_p ({\mathbb T})\) for every \(p > 1\). Hence to get Theorem 1, it suffices to show (Theorem 3): For every \(q > 2\), there exists a (compact) Helson set \(K \subseteq {\mathbb T}\) which supports a nonzero pseudo-function \(S \in A_q ({\mathbb T})\). This is the main result of the paper and its proof requires several lemmas and tools: Rudin-Shapiro polynomials, a quantitative version of a lemma of Kahane, a generalization of the Azuma inequality (which is itself a generalization of the Bernstein inequality for the sums of independent random variables), measures \(\mu_s\), \(0 < s < 1\), constructed as Riesz-type products; construction of “almost independent” random variables, exponential estimates of deviation for the empirical mean of these variables with respect to \(\mu_s\) and exponential \(L^2\) estimate.
At the end of the paper, the authors also explain how their Theorem 1 gives the corresponding version for \(L_p ({\mathbb R})\), make some remarks and ask some questions.
Reviewer: Daniel Li (Lens)

MSC:

42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
42A63 Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization
42A65 Completeness of sets of functions in one variable harmonic analysis
43A45 Spectral synthesis on groups, semigroups, etc.
43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
47A16 Cyclic vectors, hypercyclic and chaotic operators
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References:

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