IJPAM: Volume 36, No. 3 (2007)

ADDENDUM TO: ``BLOCKS OF CONSECUTIVE INTEGERS
IN SUMSETS ''

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 For $t\ge 1,$ denote by the set of the elements 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 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 $\vert A\vert+\vert B\vert=(4n+4t-5)/3.$