Generation of measures on the torus with good sequences of integers

TL;DR

This paper studies sequences of integers that generate well-defined measures on the torus, characterizing their possible limit measures for irrational rotations and exploring the extent of measures achievable through such sequences.

Contribution

It introduces the concept of good sequences on the integers, characterizes their limit measures for irrational rotations, and demonstrates the range of measures achievable using probabilistic methods.

Findings

Limit measures for good sequences are always continuous for irrational rotations.

Any measure absolutely continuous w.r.t. Lebesgue measure can be realized as a limit measure.

Some measures, like the Cantor measure, cannot be obtained as a limit measure for certain irrationals.

Abstract

Let $S= (s_1<s_2<\dots)$ be a strictly increasing sequence of positive integers and denote $\mathbf{e}(\beta)=\mathrm{e}^{2\pi i \beta}$. We say $S$ is good if for every real $\alpha$ the limit $\lim_N \frac1N\sum_{n\le N} \mathbf{e}(s_n\alpha)$ exists. By the Riesz representation theorem, a sequence $S$ is good iff for every real $\alpha$ the sequence $(s_n\alpha)$ possesses an asymptotic distribution modulo 1. Another characterization of a good sequence follows from the spectral theorem: the sequence $S$ is good iff in any probability measure preserving system $(X,\mathbf{m},T)$ the limit $\lim_N \frac1N\sum_{n\le N}f\left(T^{s_n}x\right)$ exists in $L^2$-norm for $f\in L^2(X)$. Of these three characterization of a good set, the one about limit measures is the most suitable for us, and we are interested in finding out what the limit measure $\mu_{S,\alpha}= \lim_N\frac1N\sum_{n\le N} \delta_{s_n\alpha}$ on the torus can be. In this first paper on the subject, we investigate the case of a single irrational $\alpha$. We show that if $S$ is a good set then for every irrational $\alpha$ the limit measure $\mu_{S,\alpha}$ must be a continuous Borel probability measure. Using random methods, we show that the limit measure $\mu_{S,\alpha}$ can be any measure which is absolutely continuous with respect to the Haar-Lebesgue probability measure on the torus. On the other hand, if $\nu$ is the uniform probability measure supported on the Cantor set, there are some irrational $\alpha$ so that for no good sequence $S$ can we have the limit measure $\mu_{S,\alpha}$ equal $\nu$. We leave open the question whether for any continuous Borel probability measure $\nu$ on the torus there is an irrational $\alpha$ and a good sequence $S$ so that $\mu_{S,\alpha}=\nu$.