It is known that the inequality \(\int_{-W/2}^{W/2} \big|\widehat{f}(\xi) \big|^2 \, d\xi \leq \int_{-W/2}^{W/2} \big|\widehat{|f|^*}(\xi) \big|^2 \, d\xi \) between the quadratic spectral concentration of a function and that of its decreasing rearrangement holds for any function \(f\in L^2 \) , \( \lvert {\textrm{supp}} f|=T \) , if and only if the product \(WT\) does not exceed the critical value \(\approx 0.81\) . We show that by restricting ourselves to characteristic functions we can enlarge this range up to \(WT\leq 4/3\) . Besides, we establish various properties of minimizers of the difference \(\int_{-W/2}^{W/2} |\widehat{{\mathbb{1}}_A^*}(\xi) | ^2 \, d\xi -\int_{-W/2}^{W/2} |\widehat{{\mathbb{1}}_A}(\xi) | ^2 \, d\xi \) over sets \(A\) of finite measure and prove that this difference is non-negative for all \(W,T>0\) if \(A\) is the union of two intervals. As a corollary, we obtain a sharp (up to a constant) estimate for the \(L^2 \) -norms of non-harmonic trigonometric polynomials with alternating coefficients \(\pm 1\) .
