Majorization and additive tuples in $\mathbb{Z}_2^n$

Abstract

Majorization is a fundamental tool for comparing how "spread out" the entries of two vectors are. Key majorization results were obtained for the integers by Hardy, Littlewood and Pólya and for $\mathbb{Z}_p$ by Lev. In this paper, we establish a powerful majorization theorem in $\mathbb{Z}_2^n$ that is an analogue of Lev's result in $\mathbb{Z}_p$. Our proof is based on a novel use of a compression argument that optimizes certain sums of additive representation counts. Our majorization theorem has several applications in additive combinatorics. We resolve the question: given subsets $A_1,\ldots,A_k\subseteq\mathbb{Z}_2^n$ of prescribed sizes, when is the function $r_k(A_1,\ldots,A_k)$, which counts the number of additive $k$-tuples in $A_1\times\cdots\times A_k$, maximized? We establish the corresponding minimization result and a characterization of all extremizers for $r_k(A,\ldots,A)$ when $k$ is odd. When $k=3$, this quantity is the number of Schur triples in $A$; as a special case, we recover a theorem of Samotij and Sudakov on Schur triples. We also obtain a convexity inequality and use it to prove an analogue of Pollard's theorem that strengthens and extends a well-known result of Bollobás and Leader in the $\mathbb{Z}_2^n$ setting.