Sumsets as unions of sumsets of subsets
Jordan S. Ellenberg

TL;DR
This paper demonstrates that sumsets in finite vector spaces can be decomposed into unions of smaller sumsets with bounded size, extending recent bounds on sum-free sets using advanced combinatorial and algebraic methods.
Contribution
It introduces a new decomposition technique for sumsets in finite fields, combining the Croot-Lev-Pach and Ellenberg-Gijswijt methods with Meshulam's result on low-rank matrices.
Findings
Sumsets can be expressed as unions of smaller sumsets with bounded size.
The bounds depend exponentially on the dimension, with a base less than q.
The approach generalizes recent bounds on sum-free sets.
Abstract
Let and be subsets of . We show there are subsets of and of such that is the union of and , with bounded by with . The proof relies on the method of Croot-Lev-Pach and Ellenberg-Gijswijt on the cap set problem, together with a result of Meshulam on linear spaces of low-rank matrices. The result is a modest generalization of the recent bounds on (single-colored and multi-colored) sum-free sets by the author and others.
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