Difference sets in ℝ^d

2021/10/18 by David Conlon, Conlon, David, Jeck Lim +1
Mathematics · #11B13 #11B30 #11P70 #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2110.09053

openalex publication_date 2021/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let d ≥ 2 be a natural number. We show that |A-A| ≥ (2d-2 + (1)/(d-1))|A|-(2d²-4d+3) for any sufficiently large finite subset A of ℝ^d that is not contained in a translate of a hyperplane. By a construction of Stanchescu, this is best possible and thus resolves an old question first raised by Uhrin.

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