$L^p$-Khinchin sequences and transference principles

Abstract

A sequence $(λ_n)_{n\ge1}$ of distinct positive integers is called an $L^p$-Khinchin sequence, where $1\le p\le\infty$, if $n^{-1}\sum_{k=1}^n f(λ_k x) \longrightarrow \int_{\mathbb T}f\,d\mathfrak m$ a.e. $x\in\mathbb T$ for every $f\in L^p(\mathbb T)$. When $1\le p<\infty$, we characterize the $L^p$-Khinchin sequences by a weak $(p,p)$ maximal inequality for the associated averaging operators. We also establish a transference principle reducing the weak $(p,p)$ maximal inequality to weak $(p,p)$ maximal inequalities for three model systems: translation along a lattice path in the infinite direct sum of the semigroup $\mathbb{N}_0$, translation along a lattice path in the infinite direct sum of the group $\mathbb{Z}$ and translation on the circle $\mathbb{T}$. As consequences, we prove that $(2^n+1)_{n\ge1}$, the Fibonacci sequence, sequences of distinct primes and their partial products are \emph{not} $L^p$-Khinchin for any $1\le p<\infty$. We study the special class of multiplicative sequences generated by infinite words and we show that balanced infinite words generate $L^1$-Khinchin sequences, while typical random multiplicative sequences over at least two coprime integers are almost surely not $L^p$-Khinchin for any $1\le p<\infty$. We also obtain a {Conze-type principle} for countably generated semigroup actions on a probability space.