All Questions

Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication....

Let $(X,\mu)$ be a standard probability space and let $T:X \to X$ be a measure-class preserving transformation such that there is no $T$-invariant measure absolutely continuous with respect to $\mu$. (...

Let $f : \mathbb R\to\mathbb C$ be an $1$-periodic and sufficiently smooth function, which has zero average, and let $\alpha$ irrational. We know the following: a. $\lim_{n\to\infty}\frac{1}{n}\sum_{...

Let $f:\mathbb R\to\mathbb C$ be a sufficiently smooth and $1$-periodic function of average zero (i.e., $\int_0^{1}f(x)\,dx=0$), and let $\alpha\in(0,1)\smallsetminus\mathbb Q$. We know that $$ \...