  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.
