2022/03/24 by Yifan Jing, Jing, Yifan, Akshat Mudgal +1
Mathematics · #05D40 #11B30 #11B83 #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2203.13174
openalex publication_date 2022/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given h,g ∈ ℕ, we write a set X ⊂ ℤ to be a Bh+[g] set if for any n ∈ ℤ, the number of solutions to the additive equation n = x1 + … + xh with x1, …, xh ∈ X is at most g, where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative Bh×[g] set analogously. In this paper, we prove, amongst other results, that there exist absolute constants g ∈ ℕ and δ>0 such that for any h ∈ ℕ and for any finite set A of integers, the largest Bh+[g] set B inside A and the largest Bh×[g] set C inside A satisfy max \ |B| , |C| \ ≫h |A|(1+ δ)/h . In fact, when h=2, we may set g = 31, and when h is sufficiently large, we may set g = 1 and δ≫ (log log h)1/2 - o(1). The former makes progress towards a recent conjecture of Klurman--Pohoata and quantitatively strengthens previous work of Shkredov.