Abstract and Applied Analysis

On Costas Sets and Costas Clouds

Konstantinos Drakakis

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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 2 , 2 , and 2 , 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 .

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

First available in Project Euclid: 16 March 2010

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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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  • J. P. Costas, ``Medium constraints on sonar design and performance,'' Technical Report Class 1 Rep R65EMH33, General Electric, 1965.
  • J. P. Costas, ``A study of detection waveforms having nearly ideal range-doppler ambiguity properties,'' Proceedings of the IEEE, vol. 72, no. 8, pp. 996--1009, 1984.
  • S. W. Golomb, ``Algebraic constructions for Costas arrays,'' Journal of Combinatorial Theory Series A, vol. 37, no. 1, pp. 13--21, 1984.
  • S. W. Golomb, ``Constructions and properties of Costas arrays,'' Proceedings of the IEEE, vol. 72, no. 9, pp. 1143--1163, 1984.
  • K. Drakakis and S. Rickard, ``On the generalization of the Costas property in the continuum,'' Advances in Mathematics of Communications, vol. 2, no. 2, pp. 113--130, 2008.
  • K. Drakakis, ``A review of Costas arrays,'' Journal of Applied Mathematics, vol. 2006, Article ID 26385, 32 pages, 2006.
  • W. C. Babcock, ``Intermodulation interference in radio systems/frequency of occurrence and control by channel selection,'' Bell System Technical Journal, vol. 31, pp. 63--73, 1953.
  • S. Sidon, ``Ein Satz über trigonometrische Polynome und seine Anwendung in der Theorie der Fourier-Reihen,'' Mathematische Annalen, vol. 106, no. 1, pp. 536--539, 1932.
  • A. Dimitromanolakis, Analysis of the Golomb Ruler and the Sidon set problems, and determination of large, near-optimal Golomb rulers, Diploma thesis, Department of Electronic and Computer Engineering, Technical University of Crete, Crete, Greece, 2002, http://www.cs.toronto.edu/$\sim\,\!$apostol/golomb.
  • P. Erdös and P. Turán, ``On a problem of Sidon in additive number theory, and on some related problems,'' Journal of the London Mathematical Society, vol. 16, pp. 212--215, 1941.
  • B. Lindström, ``Finding finite $B_2$-sequences faster,'' Mathematics of Computation, vol. 67, no. 223, pp. 1173--1178, 1998.
  • I. Z. Ruzsa, ``Solving a linear equation in a set of integers. I,'' Acta Arithmetica, vol. 65, no. 3, pp. 259--282, 1993.
  • R. C. Bose, ``An affine analogue of Singer's theorem,'' Journal of the Indian Mathematical Society, vol. 6, pp. 1--15, 1942.
  • R. C. Bose and S. Chowla, ``Theorems in the additive theory of numbers,'' Commentarii Mathematici Helvetici, vol. 37, pp. 141--147, 1962/1963.
  • J. Aczél and J. Dhombres, Functional Equations in Several Variables, vol. 31 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1989.
  • D. H. Hyers, ``On the stability of the linear functional equation,'' Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222--224, 1941.
  • D. H. Hyers and Th. M. Rassias, ``Approximate homomorphisms,'' Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125--153, 1992.
  • Th. M. Rassias, ``On the stability of the linear mapping in Banach spaces,'' Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297--300, 1978.
  • R. Gow, private communication, 2007.
  • I. Z. Ruzsa, ``An infinite Sidon sequence,'' Journal of Number Theory, vol. 68, no. 1, pp. 63--71, 1998.
  • M. Ajtai, J. Komlós, and E. Szemerédi, ``A dense infinite Sidon sequence,'' European Journal of Combinatorics, vol. 2, no. 1, pp. 1--11, 1981.