Distinct permutation dot products

TL;DR

Let $A,B\subset\mathbb{R}$ with $|A|=|B|=n$ and distinct entries. Define $S(\pi)=\sum_{i=1}^n a_i b_{\pi(i)}$ and $\Sigma(A,B)=\{S(\pi):\pi\in S_n\}$. The paper proves $|\Sigma(A,B)| \ge c n^3$ for some absolute $c>0$, best possible up to constants (e.g., $A=B=[n]$ attains a cubic scale by the rearrangement inequality). It introduces a supportive Halasz anticoncentration bound: for finite $D$ with $|D|=m$ and additive energies $E_k(D)$, $|\Sigma(D)| \gg_k m^{2k+1}/E_k(D)$. This bound is combined with constructing a large, low-energy increment set $D$ from $(A-A)(B-B)$ (and then from $(A-A)(B-B)+(A-A)(B-B)$) via disjoint swaps to force many distinct sums, and with geometric incidence bounds to ensure the increment set is sufficiently large. The resulting argument yields $|\Sigma(A,B)| \gg n^3$ up to polylog factors; removing the logarithm via paired-switch constructions gives the optimal cubic rate. The techniques blend additive combinatorics, geometric arguments about the permutohedron, and incidence geometry with potential extensions to matrix settings.

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 $Ω(n^{3})$ 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.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2: HPZDNPTV
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2