On the generalization of the Costas property in the continuum

TL;DR

This paper extends the Costas property to functions on the continuum, constructing bijections with applications similar to discrete Costas arrays, and explores their properties over reals and rationals.

Contribution

It introduces a continuum version of the Costas property, constructs smooth and fractal-like Costas bijections, and proposes algorithms for rational cases, linking to Artin's conjecture.

Findings

Constructed Costas bijections in the real continuum.

Developed an algorithm for Costas fractal bijections over the rationals.

Linked sequences of Welch Costas arrays to smooth Costas bijections under Artin's conjecture.

Abstract

We extend the definition of the Costas property to functions in the continuum, namely on intervals of the reals or the rationals, and argue that such functions can be used in the same applications as discrete Costas arrays. We construct Costas bijections in the real continuum within the class of piecewise continuously differentiable functions, but our attempts to construct a fractal-like Costas bijection there are successful only under slight but necessary deviations from the usual arithmetic laws. Furthermore, we are able, contingent on the validity of Artin's conjecture, to set up a limiting process according to which sequences of Welch Costas arrays converge to smooth Costas bijections over the reals. The situation over the rationals is different: there, we propose an algorithm of great generality and flexibility for the construction of a Costas fractal bijection. Its success, though, relies heavily on the enumerability of the rationals, and therefore it cannot be generalized over the reals in an obvious way.