﻿ Khintchin classes and Skew products-武汉大学数学与统计学院

Khintchin classes and Skew products

In a recent work (2022), Fan-Queffelec-Queffelec define the Khintchin class of an increasing sequence of integers {a_n} as the set of those Lebesgue-integrable functions f on the circle \mathbb{T} such that the Weyl equidistribution criterion for the sequence {a_n x} holds for almost everywhere x with f. Khinchin conjectured that for the whole set of integers \mathbb{N}, the Khintchin class is the space of all integrable functions. But it was refuted by Marstrand (1970) who even proved that not all bounded functions are included in the Khintchin class. If the Khintchin class of a set of integers is equal to L^1(\mathbb{T}), we say that the set is a Khintchin set. So, \mathbb{N} is not a Khintchin set, but {2^n} is a Khinchin set by the ergodic theorem. How to construct Khintchin sets? How to determine the Khintchin class for a given set of integers? This is a huge problem. We can also rise the problem on any compact Abelian group. Assume that we select independently 2 or 3 with probabiity 1/2 to get a random sequence \omega_1, \omega_2, \cdots. Then consider the random products w_1=\omega_1, w_2=\omega_1\omega_2, \cdots (w_n is the product of the first n selected random integers). We conjecture that {w_n} is almost surely a Khintchin set. For the moment we can not prove this conjecture, but we can prove some facts which supports this conjecture. Actually the skew product is a general way to generate random sequences, which are expected to be Khintchin sets. In this talk we will present more questions than results.