Moderate-doubling sets in $\mathbb{F}_2^n$ intersect subspaces

TL;DR

The paper analyzes sets A in the vector space $\mathbb{F}_2^n$ with moderate additive doubling $|A+A|=|A|^{2-\eta}$. It develops an entropic framework, building on the Gowers–Green–Manners–Tao program, to pass from doubling information to structural conclusions about intersection with subspaces via a sequence of endgame and fiber arguments. The main result shows there exists a subspace $V$ with $\dim V = O_{\eta}(\log|A|)$ such that the average entropy of $A$-projections onto $V$ is large, yielding a translate that intersects $A$ in at least $|A|^{\eta-\varepsilon}$ points; this improves previous dimension bounds and clarifies the subspace-structure in moderate-doubling regimes. The work further derives corollaries and highlights the entropic method as a robust tool for understanding near-extremal additive structure in $\mathbb{F}_2^n$.

Abstract

We show that any set $A$ in $\mathbb F_2^n$ with $|A+A| \le |A|^{2-η}$ must intersect a subspace of dimension $O_η(\log |A|)$ in at least $|A|^{η- o(1)}$ elements.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.2
• Corollary 1
• proof : Proof of Theorem \ref{['thm:combinatorial-corollary']}
• Corollary 2
• Theorem 2.1: Entropic Marton's Conjecture GowersGreenMannersTao2025*Theorem 1.8
• Corollary 3
• proof
• Lemma 1: Entropic Balog--Szemerédi--Gowers GowersGreenMannersTao2025*Lemma A.2