A sharp lower bound for $h$-fold restricted sumsets of integers

Abstract

Let $A\subseteq[0, L]$ be a set of $k\geqslant 9$ integers such that $0, L\in A$ and $\gcd(A)=1$. Define the $h$-fold restricted sumset $$h^{\wedge}A=\{a_1+\cdots+a_h: a_i\in A,\ 1\leqslant i\leqslant h, \ \ a_i\neq a_j\ {\rm for}\ i\neq j\}. $$ In this paper, we prove that, for every $1\leqslant h\leqslant k-1$, we have $$|h^{\wedge}A|\geqslant h(k-h)-k+2+\min\{L, h(k-h)-\mathbf{1}_h\}, $$ where $\mathbf{1}_h=1$ for $h\in\{2, k-2\}$ and $\mathbf{1}_h=0$ otherwise. The bound is sharp for every $1\leqslant h\leqslant k-1$ and the cardinality threshold $k\geqslant 9$ cannot be decreased. We also give a lower bound for the union of two consecutive restricted sumsets in a cyclic group and characterize equality.