Rotations by roots of unity and Diophantine approximation

Abstract

For a fixed integer $n$, we study the question whether at least one of the numbers $\Re X{\omega }^k$, $1\le k\le n$, is $\epsilon$-close to an integer, for any possible value of $X\in \mathbb{C}$, where $\omega$ is a primitive $n$th root of unity. It turns out that there is always a $X$ for which the above numbers are concentrated around $1/2\mathrm{mod}1$. The distance from $1/2$ depends only on the local properties of $n$, rather than its magnitude. This is directly related the so-called "pyjama" problem which was solved recently.

7 pages

Cited by in corpus (1)

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