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Recurrences in multidimensional arithmetic sequences

Published online by Cambridge University Press:  24 October 2008

S. Hemmer  and
P. C. Hemmer
S. Hemmer
Affiliation:
Blussuvoll skole and University of Trondheim
P. C. Hemmer
Affiliation:
Blussuvoll skole and University of Trondheim
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Extract

Let unity and the irrational components ri of a d-dimensional vector r be linearly independent, and consider the integers n for which

where {x} denotes the fractional part of x. It is a trivial restriction to assume that ri and φi lie between 0 and 1. This note is concerned with the question: what are the gaps between the successive n?

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 3 , May 1983 , pp. 393 - 396
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

REFERENCES

(1)Davidon, W. C. and Hemmer, P. C.The three recurrence times in irrational arithmetic sequences. Ark. Fys. Sem. Trondheim no. 5 (1977).Google Scholar
(2)Florek, K.Une remarque sur la répartition des nombres nξ, (mod 1). Colloq. Math. 2 (1951), 323–324.Google Scholar
(3)Slater, N. B.The distribution of the integers for which {ΘN} < φ. Proc. Cambridge Philos. Soc. 46 (1950), 525–534; Gaps and steps for the sequence nΘ mod 1. Proc. Cambridge Philos. Soc. 63 (1967), 1115–1123.CrossRefGoogle Scholar
(4)Weyl, H.Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77 (1916), 313–352.CrossRefGoogle Scholar