Maximizing Irregularity and the Golomb Ruler Problem

The problem of attempting to construct radar signals whose ambiguity functions are sharply peaked in both time and frequency gives rise to a quantitative concept of the irregularity of a distribution of points. The problem of maximizing the irregularity of N points distributed on a bounded closed interval is here examined. A discrete mathematical tool for studying the irregularity function is developed | Pascal-like triangles constructed from non-negative integers obeying a particular addition rule. These triangles make explicit the structure of the irregularity function; all questions about the (continuous) irregularity function become questions about this integral construction, many of which can be answered easily and elegantly. One important question about the integral construction has been previously studied under the name of the Golomb Ruler Problem. It turns out that maximizing irregularity and constructing Golomb rulers are closely related problems, each providing a interesting new perspective from which to study the other.