Khintchin conjecture and related topics

TL;DR

This paper investigates Khintchin sequences linked to integer sequences, exploring their properties, constructions, and connections to ergodic theory and group endomorphisms, advancing understanding of their role in harmonic analysis and dynamical systems.

Contribution

It introduces the concept of Khintchin sequences for group endomorphisms, establishes their basic properties, and connects their behavior to ergodic properties in dynamical systems.

Findings

Khintchin class properties are characterized.

Constructed examples of Khintchin sequences.

Established equivalence of ergodic properties under Fourier-tightness.

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.