A Hal\'asz-type theorem for permutation anticoncentration

TL;DR

This paper extends permutation anticoncentration results by introducing a new statistic for coefficient diversity, showing that increased diversity leads to significantly stronger bounds on the probability of the sum taking any single value.

Contribution

It introduces a novel statistic to measure coefficient diversity and proves that higher diversity yields polynomially faster decay in maximum point mass, improving existing bounds.

Findings

Maximum point mass decays as n^{-5/2+o(1)} for distinct coefficients

Stronger anticoncentration bounds are achieved with increased coefficient diversity

The results generalize classical Erdős–Moser theorem to permutation sums

Abstract

Given a set $A=\{a_1,\ldots,a_n\}$ of real numbers and real coefficients $b_1,\ldots,b_n$, consider the distribution of the sum obtained by pairing the $a_i$'s with the $b_i$'s according to a uniformly random permutation. A recent theorem of Pawlowski shows that as soon as the coefficients are not all equal, this distribution is always spread out at scale $n^{-1}$: no single value can occur with probability larger than $\frac{1}{2\lceil n/2\rceil + 1}$, and this bound is sharp in general. We show that stronger anticoncentration holds when the coefficients have additional diversity. We quantify the structure of the coefficient multiset by a simple statistic depending on its multiplicity profile, and prove that the maximum point mass of the permuted sum decays polynomially faster as this statistic grows. In particular, when the coefficients are all distinct we obtain a bound of $n^{-5/2+o(1)}$, which can be regarded as an analogue of a classical theorem of Erd\H{o}s and Moser.