论文标题
用良好整数序列在圆环上产生措施
Generation of measures on the torus with good sequences of integers
论文作者
论文摘要
令$ s =（s_1 <s_2 <\ dots）$是严格增加正整数的序列，并表示$ \ mathbf {e}（β）（β）= \ Mathrm {e}^{2πiβ} $。我们说$ s $如果对于每个实际$α$ lim $ \ lim_n \ frac1n \ sum_ {n \ le n} \ mathbf {e}（s_nα）$都存在。由Riesz表示定理，如果对于每个实际$α$，序列$ s $都很好。 \ frac1n \ sum_ {n \ le n} f \ left（t^{s_n} x \ right）$ in $ l^2 $ -norm in l^2（x）$。 在这三个良好集合的表征中，关于极限度量的一个最适合我们，我们有兴趣找出限制度量$μ__{s，α} = \ lim_n \ frac1n \ sum_ \ sum_ {n \ le n}δ__{s_nα} $在torus上可以使用。在有关该主题的第一篇论文中，我们研究了单个非理性$α$的情况。我们表明，如果$ s $是一个不错的集合，那么对于每个非理性$α$的限制度量$μ__{s，α} $必须是连续的borel概率度量。使用随机方法，我们表明极限度量$μ_{s，α} $可以是任何措施，相对于haar-lebesgue概率度量绝对连续。另一方面，如果$ν$是Cantor Set上支持的统一概率度量，则有一些非理性的$α$，因此对于没有好的序列$ s $，我们是否可以拥有极限度量$μ__{s，α} $等于$ $ν$。我们打开一个问题，对于圆环上的任何连续的borel概率度量$ν$是否有一个非理性的$α$和一个好的序列$ s $，因此$μ__{s，α} =ν$。
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 $μ_{S,α}=ν$.
