A distribution on triples with maximum entropy marginal

TL;DR

This paper constructs a symmetric probability distribution on triples summing to n with maximum entropy marginal, confirming a conjecture related to sum-free sets and advancing understanding of entropy-maximizing distributions.

Contribution

It provides the first explicit construction of an $S_3$-symmetric distribution with maximum entropy marginal for sum constraints, confirming a conjecture in combinatorics.

Findings

Verified the existence of the maximum entropy distribution

Constructed an explicit symmetric distribution

Confirmed a conjecture related to sum-free sets

Abstract

We construct an $S_3$-symmetric probability distribution on $\{(a,b,c) \in \mathbb{Z}_{\geq 0}^3 \: : \: a+b+c =n \}$ such that its marginal achieves the maximum entropy among all probability distributions on $\{0,1,\ldots,n\}$ with mean $n/3$. Existence of such a distribution verifies a conjecture of Kleinberg, Sawin and Speyer, which is motivated by the study of sum-free sets.