A remark on indicator functions with gaps in the spectrum

Developing a recent result of F. Nazarov and A. Olevskii, we show that for every subset $a$ of $\mathbb R$ of finite measure and every $\varepsilon>0$, there exists $b\subset\mathbb R$ with $|b|=|a|$ and $|(b\setminus a)\cup (a\setminus b)|\le\varepsilon$ such that the spectrum of $\chi_b$ is fairly thin. A generalization to locally compact Abelian groups is also provided.