On Costas Sets and Costas Clouds

Abstract

We abstract the definition of the Costas property in the context of a group and study specifically denseCostas sets (named Costas clouds) in groups with the topological property that they are dense in themselves:as a result, we prove the existence of nowhere continuous dense bijections that satisfy the Costas property on ${\mathbb{Q}}^2$, ${ℝ}^2$, and ${ℂ}^2$, the latter two being based on nonlinear solutions of Cauchy's functional equation, as well as on$\mathbb{Q}$, $ℝ$, and $ℂ$, which are, in effect, generalized Golomb rulers. We generalize the Welch and Golomb constructionmethods for Costas arrays to apply on $ℝ$ and $ℂ$, and we prove that group isomorphisms on and tensor products of Costas sets result to new Costas sets with respect to an appropriate set of distance vectors. We also give two constructive examples of a nowhere continuous function that satisfies a constrained form of the Costas property (over rational or algebraic displacements only, i.e.), based on the indicator function of a dense subset of $ℝ$.