Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Khintchin conjecture and related topics

Aihua Fan, Shilei Fan, Hervé Queffélec, Martine Queffélec

TL;DR

This work develops a comprehensive framework for Khintchin sequences beyond the classical integer setting by introducing and analyzing the Khintchin class for sequences of group endomorphisms and their skew products. It characterizes $L^p$-Khintchin sequences on the circle, explores how ordering and union/intersection operations affect the Khintchin property, and constructs new examples via multiplicative rules and Thue-Morse-type sequences. Extending to compact abelian groups, it introduces ud-sequences of epimorphisms, establishes $L^p$-convergence, and analyzes the sequence of powers and products of epimorphisms. The skew-product approach, under a Fourier-tightness condition, demonstrates that ergodicity and various mixing properties of the base system are inherited by the skew product, and it yields random Khintchin-type results in a broad, abstract setting. Together, these results illuminate when typical fibre orbits form Khintchin sequences and outline open questions in ergodic theory with potential applications to harmonic analysis on groups.

Abstract

Motivated by Khintchin's 1923 conjecture, refuted by Marstrand in 1970, we study the Khintchin class of functions associated to a given increasing sequence of integers. When the Khintchin class contains L^p(\mathbb{T}), we call the sequence a L^p-Khintchin sequence. We establish basic properties of Khintchin sequences, provide several constructions, and propose open problems for further research. We also initiate the study of Khintchin sequences of group endomorphisms on compact abelian groups. Under a Fourier-tightness assumption, we show that ergodicity (respectively, weakly mixing or strongly mixing) of a skew product of endomorphisms is equivalent to the corresponding property of the base system, supporting the idea that typical fiber orbits in such skew products should form Khintchin sequences.

Khintchin conjecture and related topics

TL;DR

This work develops a comprehensive framework for Khintchin sequences beyond the classical integer setting by introducing and analyzing the Khintchin class for sequences of group endomorphisms and their skew products. It characterizes -Khintchin sequences on the circle, explores how ordering and union/intersection operations affect the Khintchin property, and constructs new examples via multiplicative rules and Thue-Morse-type sequences. Extending to compact abelian groups, it introduces ud-sequences of epimorphisms, establishes -convergence, and analyzes the sequence of powers and products of epimorphisms. The skew-product approach, under a Fourier-tightness condition, demonstrates that ergodicity and various mixing properties of the base system are inherited by the skew product, and it yields random Khintchin-type results in a broad, abstract setting. Together, these results illuminate when typical fibre orbits form Khintchin sequences and outline open questions in ergodic theory with potential applications to harmonic analysis on groups.

Abstract

Motivated by Khintchin's 1923 conjecture, refuted by Marstrand in 1970, we study the Khintchin class of functions associated to a given increasing sequence of integers. When the Khintchin class contains L^p(\mathbb{T}), we call the sequence a L^p-Khintchin sequence. We establish basic properties of Khintchin sequences, provide several constructions, and propose open problems for further research. We also initiate the study of Khintchin sequences of group endomorphisms on compact abelian groups. Under a Fourier-tightness assumption, we show that ergodicity (respectively, weakly mixing or strongly mixing) of a skew product of endomorphisms is equivalent to the corresponding property of the base system, supporting the idea that typical fiber orbits in such skew products should form Khintchin sequences.
Paper Structure (21 sections, 28 theorems, 133 equations)

This paper contains 21 sections, 28 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Khintchin sequences of integers
  3. Characterization of $L^p$-Khintchin sequences for $1\le p\le \infty$
  4. Khintchin property depends on the ordering
  5. Union and intersection of Khintchin sequences
  6. Construction by multiplication and suggested questions
  7. Thue-Morse sequence is a Khintchin sequence
  8. Regular functions and Khintchin classes
  9. Khintchin sequences of group endomorphisms
  10. Epimorphisms on compact abelian groups
  11. Concepts of Khintchin class and Khintchin sequence
  12. $L^p$-convergence
  13. The sequence of powers of an epimorphism
  14. ud-sequences of epimorphisms and products of epimorphisms
  15. Skew products of endomorphisms on groups
  16. ...and 6 more sections

Key Result

Theorem 2.1

Assume $1\le p<\infty$. A sequence $(\lambda_n)$ of integers is a $L^p$-Khintchin sequence if and only if $T^*f(x)<\infty$$\mathbf{m}$-a.e. for every $f\in L^p(\mathbb{T})$, where $T^*f(x) = \sup_n |T_nf(x)|$ ($T_n$ being the Khintchin operator).

Theorems & Definitions (48)

  • Theorem 2.1: Characterization of $L^p$-Khintchin sequences for $1\le p<\infty$
  • proof
  • Theorem 2.2: Characterization of $L^\infty$-Khintchin sequence
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • Corollary 2.6
  • ...and 38 more