Distinct permutation dot products

TL;DR

This paper proves that for any two sets of real numbers, the sums formed by permuting one set against the other produce at least on the order of n^3 distinct values, using advanced anticoncentration techniques.

Contribution

It establishes a lower bound of Omega(n^3) on the number of distinct permutation dot products for any two real sets, introducing a supportive version of Halász's anticoncentration theorem.

Findings

Permutation sums have at least Omega(n^3) distinct values.

A new supportive anticoncentration theorem is developed.

The results apply broadly to real number sets.

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 $\sum_{i=1}^n a_i\,b_{\pi(i)}$ always take on $\Omega(n^{3})$ distinct values, as we range over all permutations $\pi \in S_n$. An important ingredient is a ``supportive'' version of Hal\'asz's anticoncentration theorem from Littlewood-Offord theory, which may be of independent interest.