On the structure of the h-fold sumsets

Jun-Yu Zhou,Quan-Hui Yang

doi:10.5802/crmath.191

Abstract

Let A be a set of nonnegative integers. Let (hA) (t) be the set of all integers in the sumset hA that have at least t representations as a sum of h elements of A. In this paper, we prove that, if k≥2, and A=a 0 ,a 1 ,⋯,a k is a finite set of integers such that 0=a 0 <a 1 <⋯<a k and gcda 1 ,a 2 ,⋯,a k =1, then there exist integers c t ,d t and sets C t ⊆[0,c t -2], D t ⊆[0,d t -2] such that

Highlights

We prove that, if k ≥ 2, and A = a0, a1, . . . , ak is a finite set of integers such that 0 = a0 < a1 < · · · < ak and gcd a1, a2, . . . , ak = 1, there exist integers ct, dt and sets Ct ⊆ [0, ct − 2], Dt ⊆ [0, dt − 2] such that (h A)(t) = Ct ∪ ct, hak − dt ∪ hak−1 − Dt for all h ≥
For h ≥ 2, we denote by h A the h-fold sumset of A, which is the set of all integers n of the form n = a1 + a2 + · · · + ah, where a1, a2, . . . , ah are elements of A and not necessarily distinct
For every positive integer t, let (h A)(t) be the set of all integers n that have at least t representations as the sum of h elements of A, that is, (h A)(t) = n ∈ Z : r A,h (n) ≥ t

Summary

Introduction

In [3, 4], Nathanson proved the following fundamental beautiful theorem on the structure of h-fold sumsets. For every positive integer h, the h-fold representation function r A,h(n) counts the number of representations of n as the sum of h elements of A. For every positive integer t , let (h A)(t) be the set of all integers n that have at least t representations as the sum of h elements of A, that is,. Nathanson [5] found that the sumsets (h A)(t) have the same structure as the sumset h A and proved the following theorem.

Some Lemmas

Proofs