## Abstract and Applied Analysis

### On Costas Sets and Costas Clouds

Konstantinos Drakakis

#### Abstract

We abstract the definition of the Costas property in the context of a group and study specifically dense Costas 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 ${\mathbb{Q}}^{2}$, ${\mathbb{R}}^{2}$, and ${\mathbb{C}}^{2}$, the latter two being based on nonlinear solutions of Cauchy's functional equation, as well as on $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$, which are, in effect, generalized Golomb rulers. We generalize the Welch and Golomb construction methods for Costas arrays to apply on $\mathbb{R}$ and $\mathbb{C}$, 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 $\mathbb{R}$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2009 (2009), Article ID 467342, 23 pages.

Dates
First available in Project Euclid: 16 March 2010

https://projecteuclid.org/euclid.aaa/1268745614

Digital Object Identifier
doi:10.1155/2009/467342

Mathematical Reviews number (MathSciNet)
MR2564001

Zentralblatt MATH identifier
1187.39030

#### Citation

Drakakis, Konstantinos. On Costas Sets and Costas Clouds. Abstr. Appl. Anal. 2009 (2009), Article ID 467342, 23 pages. doi:10.1155/2009/467342. https://projecteuclid.org/euclid.aaa/1268745614

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