On series Σc k f(kx) and Khinchin’s conjecture

Berkes, István; Weber, Michel

doi:10.1007/s11856-014-0036-0

Cited by 11 publications

(19 citation statements)

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“…., where (p r ) r≥1 is the sequence of primes. There is a gap between (8) and (9), and the problem of finding the optimal arithmetic function required for the L 2 norm convergence of (3) remains open.…”

Section: Resultsmentioning

confidence: 99%

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Convergence of series of dilated functions and spectral norms of GCD matrices

et al. 2015

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We establish a connection between the L 2 norm of sums of dilated functions whose jth Fourier coefficients are O(j −α ) for some α ∈ (1/2, 1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L 2 and for the almost everywhere convergence of series of dilated functions.

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Section: Resultsmentioning

confidence: 99%

“…In case of arithmetic criteria like in Theorem 3, Berkes and Weber [8] proved that if f satisfies (4) with complex Fourier coefficients a j , then the series (3) converges almost everywhere provided…”

Section: Resultsmentioning

confidence: 99%

Convergence of series of dilated functions and spectral norms of GCD matrices

et al. 2015

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“…We believe that Bourgain's approach goes beyond the setting explored in [1], [2], [3] and should deserve further investigations. The author has obtained in [38], [6], [33] extensions of these criteria and applied them to similar questions. He further studied in [34], [36], [37] the geometry of the sets C f defined in (3.2), as well as and their natural extension C(A) = {S n (f ), n ≥ 1, f ∈ A}, in which A is an arbitrary subset of L 2 (µ).…”

Section: Metric Entropy Criteriamentioning

confidence: 99%

“…Let us derive a criterion which has been recently applied in [6] to show the optimality of a famous theorem of Koskma. Let {h n , n ∈ Z} be a countable orthonormal basis of L 2 (µ) and use the notation f ∼ n∈Z a n (f )h n , n∈Z a 2…”

Section: Metric Entropy Criteriamentioning

confidence: 99%

Criteria of Divergence Almost Everywhere in Ergodic Theory

Weber¹

2016

J Math Sci

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Abstract. In this expository paper, we survey nowadays classical tools or criteria used in problems of convergence everywhere to build counterexamples: the Stein continuity principle, Bourgain's entropy criteria and Kakutani-Rochlin lemma, most classical device for these questions in ergodic theory. First, we state a L 1 -version of the continuity principle and give an example of its usefulness by applying it to some famous problem on divergence almost everywhere of Fourier series. Next we particularly focus on entropy criteria in L p , 2 ≤ p ≤ ∞ and provide detailed proofs. We also study the link between the associated maximal operators and the canonical Gaussian process on L 2 .

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“…Other counterexamples were given by J. Bourgain [6] by using his entropy method and by A. Quas and M. Wierdl [18]. For further results related to the Khinchin conjecture we also refer to [2] and [3] and for some generalizations we mention [1], [4] and [5].…”

Section: Introductionmentioning

confidence: 99%

Type 1 and 2 sets for series of translates of functions

et al. 2019

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Suppose Λ is a discrete infinite set of nonnegative real numbers. We say that Λ is type 1 if the series s(x) = λ∈Λ f (x + λ) satisfies a zero-one law. This means that for any non-negative measurable f : R → [0, +∞) either the convergence set C(f, Λ) = {x : s(x) < +∞} = R modulo sets of Lebesgue zero, or its complement the divergence set D(f, Λ) = {x : s(x) = +∞} = R modulo sets of measure zero. If Λ is not type 1 we say that Λ is type 2.The exact characterization of type 1 and type 2 sets is not known. In this paper we continue our study of the properties of type 1 and 2 sets. We discuss sub and supersets of type 1 and 2 sets and we give a complete and simple characterization of a subclass of dyadic type 1 sets. We discuss the existence of type 1 sets containing infinitely many algebraically independent elements. Finally, we consider unions and Minkowski sums of type 1 and 2 sets.

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On series Σc k f(kx) and Khinchin’s conjecture

Abstract: We prove the optimality of a criterion of Koksma (1953) in Khinchin's conjecture, settling a long standing open problem in analysis. Using this result, we also give a near optimal condition for the a.e. convergence of series ∞ k=1 c k f (kx) for f ∈ L 2 .

Cited by 11 publications

References 20 publications

Convergence of series of dilated functions and spectral norms of GCD matrices

Convergence of series of dilated functions and spectral norms of GCD matrices

Criteria of Divergence Almost Everywhere in Ergodic Theory

Type 1 and 2 sets for series of translates of functions

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