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Lacunary sequences in analysis, probability and number theory

Christoph Aistleitner, Istvan Berkes, Robert Tichy

Abstract

In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as equidistribution and discrepancy, metric number theory, normality, pseudorandomness, Diophantine equations, and the subsequence principle. In the final section of the paper we prove new results which provide necessary and sufficient conditions for the central limit theorem for subsequences, in the spirit of Nikishin's resonance theorem for convergence systems. More precisely, we characterize those sequences of random variables which allow to extract a subsequence satisfying a strong form of the central limit theorem.

Lacunary sequences in analysis, probability and number theory

Abstract

In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as equidistribution and discrepancy, metric number theory, normality, pseudorandomness, Diophantine equations, and the subsequence principle. In the final section of the paper we prove new results which provide necessary and sufficient conditions for the central limit theorem for subsequences, in the spirit of Nikishin's resonance theorem for convergence systems. More precisely, we characterize those sequences of random variables which allow to extract a subsequence satisfying a strong form of the central limit theorem.
Paper Structure (11 sections, 17 theorems, 152 equations)

This paper contains 11 sections, 17 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Uniform distribution and discrepancy
  3. Arithmetic effects: Diophantine equations and sums of common divisors
  4. The central limit theorem for lacunary sequences
  5. The law of the iterated logarithm for lacunary sequences
  6. Normality and pseudorandomness
  7. Random sequences
  8. The subsequence principle
  9. New results: Exact criteria for the central limit theorem for subsequences
  10. Some lemmas
  11. Proof of the theorems

Key Result

Theorem A

Let $(X_n)_{n \geq 1}$ be a determining sequence with limit random measure $\mu$ and let $A$ be a limit statute. Then there exists a subsequence $(X_{n_k})_{k \geq 1}$ such that for any further subsequence $(X_{m_k})_{k \geq 1} \subset (X_{n_k})_{k \geq 1}$ we have

Theorems & Definitions (24)

  • Theorem A: Aldous ald
  • Theorem B: Aldous ald
  • Theorem C: Berkes and Péter bp
  • Theorem D: Berkes and Péter bp
  • Definition 1
  • Theorem E: Aistleitner, Berkes and Tichy abt_perm2
  • Theorem F
  • Theorem G: Nikishin niknik2
  • Theorem H: Gaposhkin gap1972
  • Theorem J
  • ...and 14 more