The boundary structure of the sumset in Z ²

Abstract

Let A be a finite subset of Z ². Let h be a positive integer. Let hA be a sumset defined by

$$ \left\{ {h_1 a_1 + \cdot \cdot \cdot + h_k a_k |a_i \in A,\sum\limits_{i = 1}^k {h_i = h^{} } } \right\} $$

where k = |A|. It is found that the distribution of the elements of hA in the boundary region of the convex hull of hA exhibited a repeating pattern. In other words, if each side of the boundary region of the convex hull of hA is partitioned into h cells, for h sufficiently large, there exists a constant C and there exist a consecutive h − C congruent parallelograms such that the elements of hA in each parallelogram can be translated by a constant vector to obtain elements of hA in the next parallelogram. By counting the number of parallelograms and the cardinality of hA in each parallelogram, it can be found that the cardinality of hA in the boundary region is a linear function of h.

supported by PSC CUNY Grant