ACTA issues

On arithmetic progressions in a restricted sumset $A\hat{+}B$

Shu-Guang Guo, Yong-Gao Chen

Acta Sci. Math. (Szeged) 71:3-4(2005), 501-508
5884/2009

 Abstract. For any integer $a$, let $G(a)=0$ if $2|a$ and $G(a)=1$ if $2\not|a$. It is proved that for any two subsets $A$ and $B$ of $\{1, 2,\ldots, n\}$ with $|A|+|B|+G(|A|+|B|)\ge(4n+5)/3$, there exist at least $|A|+|B|+G(|A|+|B|)-3$ consecutive integers in the restricted sumset $A\hat{+} B=\{a+b : a\in A, b\in B, a\not= b\}$. Furthermore, it is proved that for any two subsets $A$ and $B$ of $\{1, 2,\ldots, n\}$ with $|A|+|B|+G(|A|+|B|)\ge(4n+4)/3$, there exists an arithmetic progression in $A\hat{+} B$ with length at least $n-1$; for any integer $r$ with $8\le r+G(r)< (4n+4)/3$, there exist two subsets $A$ and $B$ of $\{1, 2,\ldots, n\}$ with $|A|+|B|=r$ such that each arithmetic progression in $A\hat{+} B$ has length at most $2(n+1)/3$. AMS Subject Classification (1991): 11B75 Received December 30, 2002, and in revised form August 26, 2005. (Registered under 5884/2009.)