On Costas sets and Costas clouds.(English)Zbl 1187.39030

Summary: 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 $$\mathbb R$$.

MSC:

 39B22 Functional equations for real functions 39B32 Functional equations for complex functions
Full Text:

References:

 [1] J. P. Costas, “Medium constraints on sonar design and performance,” Technical Report Class 1 Rep R65EMH33, General Electric, 1965. [2] 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. [3] S. W. Golomb, “Algebraic constructions for Costas arrays,” Journal of Combinatorial Theory Series A, vol. 37, no. 1, pp. 13-21, 1984. · Zbl 0547.05020 [4] S. W. Golomb, “Constructions and properties of Costas arrays,” Proceedings of the IEEE, vol. 72, no. 9, pp. 1143-1163, 1984. · Zbl 1200.05043 [5] 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. · Zbl 1154.26300 [6] K. Drakakis, “A review of Costas arrays,” Journal of Applied Mathematics, vol. 2006, Article ID 26385, 32 pages, 2006. · Zbl 1188.94028 [7] 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. [8] 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. · Zbl 0004.21203 [9] 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/ apostol/golomb. [10] 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. · Zbl 0061.07301 [11] P. Erdös, “On a problem of Sidon in additive number theory, and on some related problems. Addendum,” Journal of the London Mathematical Society, vol. 19, p. 208, 1944. · Zbl 0061.07302 [12] B. Lindström, “Finding finite B2-sequences faster,” Mathematics of Computation, vol. 67, no. 223, pp. 1173-1178, 1998. · Zbl 0902.11013 [13] I. Z. Ruzsa, “Solving a linear equation in a set of integers. I,” Acta Arithmetica, vol. 65, no. 3, pp. 259-282, 1993. · Zbl 1042.11525 [14] R. C. Bose, “An affine analogue of Singer/s theorem,” Journal of the Indian Mathematical Society, vol. 6, pp. 1-15, 1942. · Zbl 0063.00542 [15] R. C. Bose and S. Chowla, “Theorems in the additive theory of numbers,” Commentarii Mathematici Helvetici, vol. 37, pp. 141-147, 1962/1963. · Zbl 0109.03301 [16] 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. · Zbl 0685.39006 [17] 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. · Zbl 0061.26403 [18] D. H. Hyers and Th. M. Rassias, “Approximate homomorphisms,” Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125-153, 1992. · Zbl 0806.47056 [19] 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. · Zbl 0398.47040 [20] R. Gow, private communication, 2007. [21] I. Z. Ruzsa, “An infinite Sidon sequence,” Journal of Number Theory, vol. 68, no. 1, pp. 63-71, 1998. · Zbl 0927.11005 [22] M. Ajtai, J. Komlós, and E. Szemerédi, “A dense infinite Sidon sequence,” European Journal of Combinatorics, vol. 2, no. 1, pp. 1-11, 1981. · Zbl 0474.10038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.