Generation of measures on the torus with good sequences of integers

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 $\mu_{S,\alpha}=\nu$.