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Entropic van der Corput's Difference Theorem

Weichen Gu, Xiang Li

Abstract

We prove an entropy version of van der Corput's difference theorem: the entropy of a sequence is equal to the entropy of its differences. This reveals a potential correspondence between the theory of uniform distribution mod 1 and entropy. As applications, we establish the corresponding entropy versions for several other results on uniform distribution.

Entropic van der Corput's Difference Theorem

Abstract

We prove an entropy version of van der Corput's difference theorem: the entropy of a sequence is equal to the entropy of its differences. This reveals a potential correspondence between the theory of uniform distribution mod 1 and entropy. As applications, we establish the corresponding entropy versions for several other results on uniform distribution.
Paper Structure (9 sections, 24 theorems, 140 equations, 1 table)

This paper contains 9 sections, 24 theorems, 140 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries on anqie entropy of sequences
  3. Entropic van der Corput’s theorem
  4. A special case: entropy of $p^n x \mod 1$
  5. Dual entropy
  6. Properties of dual entropy
  7. A generalization of Furstenberg's $\times 2\times 3$ conjecture
  8. Further discussions on the Möbius disjointness conjecture
  9. Proof of Lemma \ref{['231210lem1']}

Key Result

Theorem 1

Let $\{x(n)\}_{n\in\mathbb N}$ be a sequence taking values in the torus $\mathbb R/\mathbb Z$. Assume that for every $d \ge1$, the sequence $\{x(n+d)-x(n)\}_{n\in\mathbb N}$ is uniformly distributed. Then $\{x(n)\}_{n\in \mathbb N}$ is uniformly distributed.

Theorems & Definitions (40)

  • Theorem : Van der Corput's Difference Theorem, van1931diophantische
  • Theorem 1.1: Entropic van der Corput's Theorem
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Anqie, ge2016wei2022anqie
  • Definition 2.2: Anqie entropy, ge2016
  • Lemma 2.3: ge2016
  • Lemma 2.4: ge2016wei2022anqie
  • Lemma 2.5: ge2016
  • ...and 30 more