  It is known that the inequality
\begin{align*}\int_{-W/2}^{W/2}|\widehat{f}(\xi)|^2d\xi\leq
\int_{-W/2}^{W/2}|\widehat{|f|^*}(\xi)|^2d\xi \end{align*} between the
quadratic spectral concentration of a function and that of its decreasing
rearrangement holds for any function $f\in L^2,\;|\text{supp} f|=T,$ if and
only if the product $WT$ does not exceed the critical value $\approx 0.884$. 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
$\int_{-W/2}^{W/2}|\widehat{\chi_A^*}(\xi)|^2d\xi-\int_{-W/2}^{W/2}|\widehat{\chi_A}(\xi)|^2d\xi$
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 $\pm 1$.
