Developing a recent result of F. Nazarov and A. Olevskii, we show that for every subset a in ℝ of finite measure and every ε > 0 there exists b ⊂ ℝ with |b| = |a| and |(b \ a) ∪ (a \ b)|≤ε such that the spectrum of χ_b is fairly thin. A generalization to locally compact Abelian groups is also provided.