Additively stable sets, critical sets for the 3k-4 theorem in ℤ and ℝ

2023/11/23 by Péringuey, Paul, de Roton, Anne
#11P70 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2311.13891

Abstract

We describe in this paper additively left stable sets, i.e. sets satisfying ((A+A)-inf(A))∩[inf(A),sup(A)]=A (meaning that A-inf(A) is stable by addition with itself on its convex hull), when A is a finite subset of integers and when A is a bounded subset of real numbers. More precisely we give a sharp upper bound for the density of A in [inf(A),x] for x≤sup(A), and construct sets reaching this density for any given x in this range. This gives some information on sets involved in the structural description of some critical sets in Freiman's 3k-4 theorem in both cases.

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