Planar additive bases for rectangles

Abstract

We study a generalization of additive bases into a planar setting. A planar additive basis is a set of non-negative integer pairs whose vector sumset covers a given rectangle. Such bases find applications in active sensor arrays used in, for example, radar and medical imaging. The problem of minimizing the basis cardinality has not been addressed before. We propose two algorithms for finding the minimal bases of small rectangles: one in the setting where the basis elements can be anywhere in the rectangle, and another in the restricted setting, where the elements are confined to the lower left quadrant. We present numerical results from such searches, including the minimal cardinalities for all rectangles up to $[0,11]\times[0,11]$, and up to $[0,46]\times[0,46]$ in the restricted setting. We also prove asymptotic upper and lower bounds on the minimal basis cardinality for large rectangles.