On nowhere continuous Costas functions and infinite Golomb rulers

TL;DR

This paper demonstrates the existence of nowhere continuous bijections with the Costas property and infinite Golomb rulers, introduces dense Costas clouds, and provides constructive examples of constrained Costas functions based on nonlinear functional solutions.

Contribution

It establishes the existence of nowhere continuous Costas functions and Golomb rulers, introduces dense Costas clouds, and offers constructive examples with constrained Costas properties.

Findings

Existence of nowhere continuous Costas bijections.

Existence of dense Costas clouds in the real plane.

Constructive examples of constrained Costas functions.

Abstract

We prove the existence of nowhere continuous bijections that satisfy the Costas property, as well as (countably and uncountably) infinite Golomb rulers. We define and prove the existence of real and rational Costas clouds, namely nowhere continuous Costas injections whose graphs are everywhere dense in a region of the real plane, based on nonlinear solutions of Cauchy's functional equation. We also give 2 constructive examples of a nowhere continuous function, that satisfies a constrained form of the Costas property (over rational or algebraic displacements only, that is), based on the indicator function of a dense subset of the reals.