Generation of measures on the torus with good sequences of integers
Let $S= (s_1<s_2<\dots)$ be a strictly increasing sequence of positive integers and denote $\mathbf{e}(β)=\mathrm{e}^{2πi β}$. We say $S$ is good if for every real $α$ the limit $\lim_N \frac1N\sum_{n\le N} \mathbf{e}(s_nα)$ exists. By the Riesz representation theorem, a sequence $S$ is good iff for every real $α$ the sequence $(s_nα)$ 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 $μ_{S,α}= \lim_N\frac1N\sum_{n\le N} δ_{s_nα}$ on the torus can be. In this first paper on the subject, we investigate the case of a single irrational $α$. We show that if $S$ is a good set then for every irrational $α$ the limit measure $μ_{S,α}$ must be a continuous Borel probability measure. Using random methods, we show that the limit measure $μ_{S,α}$ can be any measure which is absolutely continuous with respect to the Haar-Lebesgue probability measure on the torus. On the other hand, if $ν$ is the uniform probability measure supported on the Cantor set, there are some irrational $α$ so that for no good sequence $S$ can we have the limit measure $μ_{S,α}$ equal $ν$. We leave open the question whether for any continuous Borel probability measure $ν$ on the torus there is an irrational $α$ and a good sequence $S$ so that $μ_{S,α}=ν$.