On the number of permutation-twisted dot products

TL;DR

This paper proves that the sum of permutation-twisted dot products over a field of characteristic zero takes at least on the order of n^3 distinct values, advancing understanding of their combinatorial and anticoncentration properties.

Contribution

It establishes a lower bound of (n^3) on the number of distinct sums for permutation-twisted dot products, which is optimal up to constants, complementing recent probabilistic anticoncentration results.

Findings

Sum assumes at least (n^3) distinct values

Supports are optimal up to constants

Advances understanding of permutation-twisted dot product properties

Abstract

Let $\mathbb{K}$ be a field of characteristic $0$. For each choice of distinct $a_1, \ldots, a_n\in \mathbb{K}$ and distinct $b_1, \ldots, b_n\in \mathbb{K}$, consider the sum $S=\sum_{i=1}^n a_i b_{\pi(i)}$ as $\pi$ ranges over the permutations of $[n]$. We show that this sum always assumes at least $\Omega(n^3)$ distinct values. This ``support'' bound, which is optimal up to the value of the implicit constant, complements recent work of Do, Nguyen, Phan, Tran, and Vu, and of Hunter, Pohoata, and Zhu on the anticoncentration properties of $S$ when $a_1,\ldots,a_n,b_1,\ldots,b_n$ are real and $\pi$ is chosen uniformly at random.