Rotations by roots of unity and Diophantine approximation

TL;DR

This paper investigates how roots of unity influence the approximation of complex numbers by integers, revealing concentration phenomena around 1/2 mod 1 and connecting to the recent solution of the pyjama problem.

Contribution

It establishes that for any fixed integer n, there exists a complex number X making certain rotations by roots of unity concentrate near 1/2 mod 1, linking to the pyjama problem.

Findings

Existence of X concentrating around 1/2 mod 1

Distance depends on local properties of n

Connection to the pyjama problem

Abstract

For a fixed integer $n$, we study the question whether at least one of the numbers $\Re X\omega^k$, $1\leq k\leq n$, is $\varepsilon$-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 $1/2\bmod1$. 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.