IJPAM: Volume 36, No. 3 (2007)

IN SUMSETS $(A+B)_t$''

Shu-Guang Guo
Department of Mathematics
Yancheng Teachers College
Yancheng, Jiangsu, 224002, P.R. CHINA
e-mail: ychgsg@yahoo.com.cn

Abstract.Let $A, B\subseteq \{1, \cdots, n\}.$ For $m\in \mathbf{Z},$ let $r_{A,B}(m)$ be the cardinality of the set of ordered pairs $(a, b)\in A\times B$ such that $a+b=m.$ For $t\ge 1,$ denote by $(A+B)_t$ the set of the elements $m$ for which $r_{A,B}(m)\ge t.$ Recently Guo and Chen proved that for any subsets $A, B\subseteq \{1, \cdots, n\}$ such that $\vert A\vert+\vert B\vert\ge
(4n+4t-4)/3,$ the sumset $(A+B)_t$ contains a block of consecutive integers with the length at least $\vert A\vert+\vert B\vert-2t+1$ unless $3\vert(n+t-1)$ and $A=B=[1, (n+t-1)/3\,]\cup [\,(2n-t+4)/3, n\,].$ In this paper we give an addendum to this result by dealing with the case when $\vert A\vert+\vert B\vert=(4n+4t-5)/3.$

Received: May 30, 2006

AMS Subject Classification: 11B75, 05D05

Key Words and Phrases: additive number theory, sumsets, restricted sumsets

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 36
Issue: 3