Embeddings of circular orbits and the distribution of fractional parts

Abstract:

Let $r_{n,\alpha } (i,t)$ be the number of points of the sequence $\{t\}, \{\alpha +t\}, \{2\alpha +t\},\dots$ that fall into the semiopen interval $[0, \{n\alpha \})$, where $\{x\}$ is the fractional part of $x$, $n$ is an arbitrary integer, and $t$ is any fixed number. Denote by $\delta _{n,\alpha }(i,t)= i \{n \alpha \} - r_{n,\alpha } (i,t)$ the deviation of the expected number $i \{n \alpha \}$ of hits of the above sequence in the semiopen interval $[0, \{n\alpha \})$ of length $\{n \alpha \}$ from the observed number of hits $r_{n,\alpha } (i,t)$. E. Hecke proved the following theorem: the deviations $\delta _{n,\alpha }(i,t)$ satisfy the inequality $|\delta _{n,\alpha }(i,t)|\le |n|$ for all $t\in [0,1)$ and $i=0,1,2,\dots$. In this paper, conditions on the parameters $n$ and $\alpha$ are found under which $\delta _{n,\alpha }(i, t)$ can be bounded as $|\delta _{n,\alpha }(i, t)|< c_{\alpha }$ for a constant $c_{\alpha }>0$ depending on $\alpha$, as $|n| \rightarrow \infty$ and $n$ ranges over an infinite subset of integers. In the case where $n$ is taken to be equal to the denominators of the convergents $Q_m$ to $\alpha$, the smallest values of the constants $c_{\alpha }$ are computed. The proofs involve a new method based on embeddings of circular orbits into partitions of the unit circle.