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 , , and , the latter two being based on nonlinear solutions of Cauchy's functional equation, as well as on , , and , which are, in effect, generalized Golomb rulers. We generalize the Welch and Golomb construction methods for Costas arrays to apply on and , 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 .
Highlights
Costas arrays are square arrangements of dots and blanks such that there is exactly one dot per row and column i.e., permutation arrays, and such that no four dots form a parallelogram and no three dots on the same straight line are equidistant
Is there a function satisfying the Costas property whose graph is everywhere dense on a region of the real plane? This leads to the notion of Costas clouds, studied below
We did not further pursue the direction of the study of Costas sets over nonAbelian groups, which we leave as future work, but turned our attention to groups with the analytic property of being dense in themselves instead such as Q, R, and C : after a brief overview of the known construction methods for Costas arrays and Golomb rulers, we embarked on the study of Costas sets that are dense in the group they belong to; we named these sets Costas clouds
Summary
Costas arrays are square arrangements of dots and blanks such that there is exactly one dot per row and column i.e., permutation arrays , and such that no four dots form a parallelogram and no three dots on the same straight line are equidistant. After recognizing that some parts of the definition of a Costas array are additional, peripheral requirements, imposed for convenience by the nature of the engineering application, but not essential even for the application itself , we show that Costas arrays and Golomb rulers are essentially instantiations of the same concept/property over different groups. Though this approach opens the door for the study of “exotic” Costas structures on arbitrary algebraic groups possibly nonAbelian , we restrict our attention almost immediately on groups with the analytic property of being dense in themselves, and in particular the fields Q, R, and C. We construct explicitly two examples of nowhere continuous Costas bijections, using nowhere continuous indicator functions of dense subsets of the real line as building blocks
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