A Halász-type theorem for permutation anticoncentration

TL;DR

This work studies anticoncentration for permutation-based sums $\sum_{i=1}^n a_i b_{\pi(i)}$, extending Pawlowski's bound by introducing the multiplicity profile $\mu(B)$ and the derived statistic $M(B)$ to quantify coefficient diversity. The authors prove a Halász-type bound stating that the maximum point mass $Q$ decays like $n^{-(1-o(1))}$ scaled by $1/\sqrt{M(B)}$, and obtain a sharper bound $Q \lesssim n^{1/2}(\log n)^2/\sqrt{M(A)M(B)}$ when $M(A)M(B) \ge n^{3+\varepsilon}$, highlighting an Erdős–Moser analogue in the permutation model. The proofs combine additive-structure energy methods, particularly the Roche-Newton–Rudnev Minkowski-distance energy bound, with a Halász-type decomposition and a dyadic partitioning argument to control the anticoncentration via higher-order energy quantities $\kappa_c(A,B)$ and $\kappa'_c(A,B)$. The results are tight up to subpolynomial factors and open several avenues, including removing the $n^{o(1)}$ loss and achieving precise constants in the distinct-coefficient regime.

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 Erdős and Moser.

Theorems & Definitions (8)

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• proof : Proof of \ref{['thm:MAMB']}
• proof : Proof of \ref{['thm:main']}