Restricted sumsets in $\mathbb{Z}$

Abstract

Let $k⩾3$ and let $A=\{0=a_0<a_1<\cdots <a_{k-1}\}$ with $\gcd (A)=1$. Freiman-Lev conjecture [V.F. Lev, Restricted set addition in groups, I. The classical setting, J. London Math. Soc. 62(2000), 27-40] is a well-known conjecture which related to restricted sumsets. Up to now, Freiman-Lev conjecture is open for all $a_{k-1}⩾2k-2$. In this paper, we prove the Freiman-Lev conjecture is true for $a_{k-1}⩾2k-2$ and $a_{k-2}<2k-4$. That is, Freiman-Lev conjecture is still open for the case $a_{k-1}⩾2k-2$ and $a_{k-2}\ge 2k-4$.

34 pages