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Beyond Beatty sequences: Complementary lattices

Part of: Designs and configurations Combinatorics

Published online by Cambridge University Press:  06 August 2020

Sam Vandervelde
Sam Vandervelde*
Affiliation:
Proof School, 973 Mission St, San Francisco, CA94103
*
e-mail: [email protected]
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Abstract

By taking square lattices as a two-dimensional analogue to Beatty sequences, we are motivated to define and explore the notion of complementary lattices. In particular, we present a continuous one-parameter family of complementary lattices. This main result then yields several novel examples of complementary sequences, along with a geometric proof of the fundamental property of Beatty sequences.

Keywords

Beatty sequencecomplementary sequencesquare lattice

MSC classification

Primary: 05A17: Partitions of integers
Secondary: 05B35: Matroids, geometric lattices
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 3 , September 2021 , pp. 499 - 511
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Copyright
© Canadian Mathematical Society 2020

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References

Beatty, S., Problems and solutions: Problem 3173 . Am. Math. Month. 33(1926), 159.CrossRefGoogle Scholar
Berstel, J., Recent results on extensions of Sturmian words. Int. J. Algebra Comput. 12(2002), 371–385. http://dx.doi.org/10.1142/S2181967020095XCrossRefGoogle Scholar
Carlitz, L., Scoville, R., and Hoggatt, V. E. Jr., Lucas representations. Fibonacci Quart. 10(1972), no. 1, 29–42.Google Scholar
Fraenkel, A. S. and Krieger, D., The structure of complementary sets of integers: A 3-shift theorem. Int. J. Pure Appl. Math. 10(2004), 1–49.Google Scholar
Hashimoto, Y., A geometric approach to Beatty sequences in higher dimensions. Bull. Aichi Univ. Educ. Nat. Sci. 60(2011), 13–21.Google Scholar
Honsberger, R., Ingenuity in mathematics. Random House, New York, NY, 1970.CrossRefGoogle Scholar
Lambek, J. and Moser, L., Inverse and complementary sequences of natural numbers. Am. Math. Monthly 61(1954), no. 7, 454–458. http://dx.doi.org/10.2307/2308078CrossRefGoogle Scholar
Morse, M. and Hedlund, G. A., Symbolic dynamics II: Sturmian sequences. Am. J. Math. 61(1940), 1–42. http://dx.doi.org/10.2307/2371431CrossRefGoogle Scholar
Stolarsky, K. B., Beatty sequences, continued fractions, and certain shift operators. Canad. Math. Bull. 19(1976), no. 4, 473–482. http://dx.doi.org/10.4153/CMB-1976-071-6CrossRefGoogle Scholar
Uspensky, J. V., On a problem arising out of the theory of a certain game. Am. Math. Monthly 34(1927), 516–521. http://dx.doi.org/10.2307/2299838CrossRefGoogle Scholar