Counting the Number of Elements in h~((r))A: A Special Case

摘要

Let A = {0, 1, …, k - 2, k - 1 + b} = [0，k - 2] U {k - 1 + b} be a finite set，where b is a nonnegative integer. For h > 1, let hA be the regular h-fold sumset of A. Nathanson proved that |hA| is a strictly increasing piecewise-linear function of b for 0 < b ≤ (h- 1)(k-2) and that|hA| is constant for b > (h-1)(k-2). In this paper, we generalize the result of Nathanson by studying the cardinality of the general h-fold sumset h(r) A (r is a positive integer) which also gives the cardinality of the restricted h-fold sumset hA.