On the number of permutation-twisted dot products

TL;DR

We bound the number of distinct permutation-twisted dot products $S(\mathbf{a},\mathbf{b};\pi)$ from below by $|\mathcal{S}(\mathbf{a},\mathbf{b})|\ge c n^3$ for distinct entries in $\mathbf{a}$ and $\mathbf{b}$. The proof reduces to analyzing subset sums of products $s(i)=(a_{k_i}-a_{j_i})(b_{k_i}-b_{j_i})$ coming from a greedy, linear-sized family of disjoint transpositions, and hinges on showing that, for superadditive sequences, the set of subset sums of $s(i)$ grows like $\Theta(m^3)$. A central lemma provides a lower bound $f(m)\ge\kappa m^3$ for the minimal number of distinct subset sums, via a careful partitioning argument and a recursive bound, which together imply the main cubic growth. The technique gives sharp results up to constants and connects to anticoncentration results for random $\pi$, with potential generalizations to higher-order permutation-twisted dot products.

Abstract

For distinct real numbers $a_1, \ldots, a_n$ and distinct real numbers $b_1, \ldots, b_n$, consider the sum $S=\sum_{i=1}^n a_i b_{π(i)}$ as $π$ ranges over the permutations of $[n]$. We show that this sum always assumes at least $Ω(n^3)$ distinct values, which is optimal. This ``support'' bound complements recent work of Do, Nguyen, Phan, Tran, and Vu, and of Hunter, Pohoata, and Zhu on the anticoncentration properties of $S$ when $π$ is chosen uniformly at random.

Theorems & Definitions (6)

• Theorem 1.1
• Lemma 2.1: Superadditivity
• proof
• Lemma 2.2
• proof
• Remark 2.3