On an inverse problem for restricted sumsets

Xin-Qi Luo; Zhi-Wei Sun

paper

On an inverse problem for restricted sumsets

arXiv:2305.11574

Abstract

Let $n$ be a positive integer, and let $A$ be a set of $k \geq 2 n - 1$ integers. For the restricted sumset $S_{n} (A) = {a_{1} + \dots + a_{n} : a_{1}, \dots, a_{n} \in A, and a_{i}^{2} \neq = a_{j}^{2} for 1 \leq i < j \leq n},$ by a 2002 result of Liu and Sun we have $∣ S_{n} (A) ∣ \geq (k - 1) n - \frac{3}{2} n (n - 1) + 1.$ In this paper, we determine the structure of $A$ when the lower bound is attained.

18 pages