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A quantitative improvement on the hypergraph Balog-Szemerédi-Gowers theorem

Hyunwoo Lee

TL;DR

This work addresses the hypergraph Balog–Szemerédi–Gowers problem in abelian groups by proving a quantitative hypergraph BSG theorem with bounds that scale linearly in the density parameter $K$ and have reduced dependence on the uniformity degree $r$. The authors introduce the $r$-octopus as a hypergraph analogue of a length-3 path and develop a framework to extract large vertex subsets where every crossing $r$-tuple supports many octopuses, turning octopus configurations into many representations of restricted sumsets. They obtain explicit bounds: for $|E(H)| \ge |A_1|\cdots|A_r|/K$ and $|\bigoplus_H(A_1,\dots,A_r)| \le C(\prod_i|A_i|)^{1/r}$, there exist $A_i'\subseteq A_i$ with $|A_i'| \ge |A_i|/2^{i+2}K$ and $|A_1'+\cdots+ A_r'| \le 8^{r^3}(r-1)^{(r-1)}K^{(r^2+5r-4)/2}C^{2r-1}(\prod_i|A_i|)^{1/r}$. An almost-all version is also established via Shao’s approach and arithmetic removal. The results significantly improve previous bounds and extend the hypergraph BSG toolkit to asymmetric, non-symmetric, and dense hypergraphs, with potential applications to multi-fold sumset problems in additive combinatorics.

Abstract

In this note, we obtain a quantitative improvement on the hypergraph variant of the Balog-Szemerédi-Gowers theorem due to Sudakov, Szemerédi, and Vu [Duke Math. J.129.1 (2005): 129--155]. Additionally, we prove the hypergraph variant of the ``almost all'' version of Balog-Szemerédi-Gowers theorem.

A quantitative improvement on the hypergraph Balog-Szemerédi-Gowers theorem

TL;DR

This work addresses the hypergraph Balog–Szemerédi–Gowers problem in abelian groups by proving a quantitative hypergraph BSG theorem with bounds that scale linearly in the density parameter and have reduced dependence on the uniformity degree . The authors introduce the -octopus as a hypergraph analogue of a length-3 path and develop a framework to extract large vertex subsets where every crossing -tuple supports many octopuses, turning octopus configurations into many representations of restricted sumsets. They obtain explicit bounds: for and , there exist with and . An almost-all version is also established via Shao’s approach and arithmetic removal. The results significantly improve previous bounds and extend the hypergraph BSG toolkit to asymmetric, non-symmetric, and dense hypergraphs, with potential applications to multi-fold sumset problems in additive combinatorics.

Abstract

In this note, we obtain a quantitative improvement on the hypergraph variant of the Balog-Szemerédi-Gowers theorem due to Sudakov, Szemerédi, and Vu [Duke Math. J.129.1 (2005): 129--155]. Additionally, we prove the hypergraph variant of the ``almost all'' version of Balog-Szemerédi-Gowers theorem.
Paper Structure (6 sections, 9 theorems, 15 equations)

This paper contains 6 sections, 9 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Remark $1$.
  3. Remark $2$.
  4. Proof overview
  5. Large sets supporting many octopuses
  6. General case

Key Result

Theorem 1.1

Let $\mathbf{G}$ be an abelian group and $A \subseteq \mathbf{G}$. If $A$ has an additive energy at least $\frac{|A|^3}{C}$ for some constant $C > 0$, then there is a subset $A'\subseteq A$ with $\mathopen{}\mathclose{\left\lvert A'\right\rvert \geq \Omega_C(1) \mathopen{}\mathclose{\left\lvert A}\r

Theorems & Definitions (14)

  • Theorem 1.1: Balog-Szemerédi-Gowers balog-szemerediGowers-BSG
  • Theorem 1.2: Sudakov--Szemerédi--Vu Sudakov-Szemeredi-Vu, Tao-Vu Tao-Vu-book
  • Theorem 1.3: Sudakov-Szemerédi-Vu Sudakov-Szemeredi-Vu
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6: Mudgal MudgalMudgal-private
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4: DRC
  • ...and 4 more