Sets in ℤ^k with doubling 2^k+δ are near convex progressions

2020/04/15 by Peter van Hintum, Hunter Spink, van Hintum, Peter +3
Mathematics · #11P70 (Primary) #52A40 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2004.07264

openalex publication_date 2020/04/15 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

For δ>0 sufficiently small and A⊂ ℤ^k with |A+A|≤ (2^k+δ)|A|, we show either A is covered by m_k(δ) parallel hyperplanes, or satisfies |\widehatco(A)∖ A|≤ c_kδ|A|, where \widehatco(A) is the smallest convex progression (convex set intersected with a sublattice) containing A. This generalizes the Freiman-Bilu 2^k theorem, Freiman's 3|A|-4 theorem, and recent sharp stability results of the present authors for sumsets in ℝ^k conjectured by Figalli and Jerison.

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