Planar additive bases for rectangles

TL;DR

This paper explores planar additive bases for rectangles, proposing algorithms to find minimal bases, providing numerical results for small rectangles, and establishing bounds for large ones, with applications in sensor array design.

Contribution

It introduces the first algorithms for minimal planar additive bases of rectangles and provides comprehensive numerical and theoretical bounds.

Findings

Minimal bases for rectangles up to 11x11 and 46x46 are computed.

Two algorithms are developed for different basis element placement constraints.

Asymptotic bounds on basis size are established for large 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.