Improved stability for the size and structure of sumsets

2024/06/05 by Granville, Andrew, Smith, Jack, Walker, Aled
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2406.03275

Abstract

Let A ⊂ ℤ^d be a finite set. It is known that the sumset NA has predictable size (\vert NA\vert = P_A(N) for some P_A(X) ∈ ℚ[X]) and structure (all of the lattice points in some finite cone other than all of the lattice points in a finite collection of exceptional subcones), once N is larger than some threshold. In previous work, joint with Shakan, the first and third named authors established the first effective bounds for both of these thresholds for an arbitrary set A. In this article we substantially improve each of these bounds, coming much closer to the corresponding lower bounds known.

Related

• Effective results on the size and structure of sumsets
• A problem on sumset sizes of sets of lattice points
• Compression and complexity for sumset sizes in additive number theory
• Explicit sumset sizes in additive number theory
• The structure of higher sumsets