2023/07/06 by Yifan Jing, Jing, Yifan, Akshat Mudgal +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2307.03066
openalex publication_date 2023/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a connected compact group equipped with the normalised Haar measure μ. Our first result shows that given α, β>0, there is a constant c = c(α,β)>0 such that for any compact sets A,B⊆ G with αμ(B)≥μ(A)≥ μ(B) and μ(A)+μ(B)≤ 1-β, there exist b1,… bc∈ B such that μ(A⋅ \b1,…,bc\)≥ μ(A)+μ(B). A special case of this, that is, when G=\mathbbTd, confirms a recent conjecture of Bollobás, Leader and Tiba. We also prove a quantitatively stronger version of such a result in the discrete setting of ℝd. Thus, given d ∈ ℕ, we show that there exists c = c(d) >0 such that for any finite, non-empty set A ⊆ ℝd which is not contained in a translate of a hyperplane, one can find a1, …, ac ∈ A satisfying |A+ \a1, …, ac\| ≥ (d+1)|A| - Od(1). The main term here is optimal and recovers the bounds given by Freiman's lemma up to the Od(1) error term.