Generalised Beatty sets

Marc Technau
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 2, Pages 127-135
DOI: 10.7546/nntdm.2019.25.2.127-135
Full paper (PDF, 196 Kb)

Details

Authors and affiliations

Marc Technau
Institute of Analysis and Number Theory
Graz University of Technology, Kopernikusgasse 24, 8010 Graz, Austria

Abstract

Generalised Beatty sets, that is, sets of the form {⌊mα₁ + nα₂ + β⌋ : m, n ∈ ℕ}, are studied, where ⌊ξ⌋ denotes the largest integer less than or equal to ξ. Such sets are shown to be contained in a suitable ordinary Beatty set {⌊nα + β⌋ : n ∈ ℕ} and equal said set save for finitely many exceptions. Moreover, bounds for the largest such exception are given.

Keywords

• Beatty sequence
• Beatty set

2010 Mathematics Subject Classification

• Primary: 11B83
• Secondary: 11K60

References

1. Banks, W. D., & Shparlinski, I. E. (2009). Prime numbers with Beatty sequences, Colloq. Math., 115 (2), 147–157.
2. Beatty, S. (1926). Problem 3173, Amer. Math. Monthly, 33, 159.
3. Beatty, S., Ostrowski, A., Hyslop, J., & Aitken, A. C. (1927). Solutions to problem 3173, Amer. Math. Monthly, 34, 159–160.
4. Bernoulli, J. III (1772). Sur une nouvelle espece de calcul, InRecueil pour les Astronomes, Volume I, 255–284, Berlin.
5. Christoffel, E. B. (1873). Observatio arithmetica, Annali di Mat., 6 (2), 148–153.
6. Christoffel, E. B. (1887). Lehrs atze uber arithmetische Eigenschaften der Irrationalzahlen, Annali di Mat. (2), 15, 253–276.
7. Hardy, G. H., & Wright, E. M. (2008). An introduction to the theory of numbers, Sixth edition, Oxford University Press, Oxford.
8. Kuipers, L., & Niederreiter, H. (1974). Uniform distribution of sequences, John Wiley & Sons, New York.
9. Perron, O. (1954). Die Lehre von den Kettenbruchen. Band I, third edition, B.G. Teubner Verlagsgesellschaft, Leipzig.
10. Steuding, J., & Technau, M. (2016). The least prime number in a Beatty sequence, J. Number Theory, 169, 144–159.
11. Strutt, J. W. (1926). The Theory of Sound, second edition, Macmillan, London.
12. Vaughan, R. C. (1978). On the distribution of αp modulo 1, Mathematika, 24, 135–14