Distinct permutation dot products

Abstract: We show that for any two sets of reals numbers $A={a_1,\dots,a_n}$ and $B={b_1,\dots,b_n}$, the sums of the form $\sum_{i=1}ⁿ a_i\,b_{π(i)}$ always take on $Ω(n<sup>{3})$ distinct values, as we range over all permutations $π\in S_n$. An important ingredient is a ``supportive'' version of Halász's anticoncentration theorem from Littlewood-Offord theory, which may be of independent interest.