Hostname: page-component-7dd5485656-j7khd Total loading time: 0 Render date: 2025-10-30T23:38:47.568Z Has data issue: false hasContentIssue false
  • English
  • Français

Gaps and steps for the sequence nθ mod 1

Published online by Cambridge University Press:  24 October 2008

Noel B. Slater
Noel B. Slater
Affiliation:
Department of Applied Mathematics, University of Hull
Get access Rights & Permissions [Opens in a new window]

Extract

For a given θ(0 < θ < 1) and r = 0, 1, 2, 3,… there are two related problems on the fractional parts {rθ}, namely:

the gap problem: for any φ (0 < φ < 1) to determine the gaps between the successive r for which

the step problem: for the set

rearranged in ascending order, to determine the steps into which the interval [0,1] is thereby partitioned. There are subsidiary problems of the numbers of gaps or steps of the various lengths, and also of the permutation of 1,…, N when (2) is arranged in order.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 4 , October 1967 , pp. 1115 - 1123
Copyright
Copyright © Cambridge Philosophical Society 1967

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

(1)Florek, K.Une remarque sur la répartition des nombres nξ (mod 1). Colloquium Math. Wroclaw, 2 (1951), 323–324.Google Scholar
(2)Halton, J. H.The distribution of the sequence {nξ} (n = 0, 1, 2, …). Proc. Cambridge Philos. Soc. 61 (1965), 665–670.CrossRefGoogle Scholar
(3)Hartman, S.Über die Abstände von Punkten nξ auf der Kreisperipherie. Ann. Soc. Polon. Math. 25 (1952), 110–114.Google Scholar
(4)Slater, N. B.The distribution of the integers N for which {θN} < φ. Proc. Cambridge Philos. Soc. 46 (1950), 525–534.CrossRefGoogle Scholar
(5)Slater, N. B.Distribution problems and physical applications. Compositio Math. 16 (1964), 176–183.Google Scholar
(6)Sós, V. T.On the theory of Diophantine approximations I. Acta Math. Acad. Sci. Hungar. 8 (1957), 461–472.CrossRefGoogle Scholar
(7)Sós, V. T.On the distribution mod 1 of the sequence nα. Ann. Univ. Sci. Budapest Eötvös Sect. Math. 1 (1958), 127–134.Google Scholar
(8)Surányi, J.Üer die Anordnung der Vielfachen einer reellen Zahl mod 1. Ann. Univ. Sci. Budapest Eötvös Sect. Math. 1 (1958), 107–111.Google Scholar
(9)Świerczkowski, S.On successive settings of an arc on the circumference of a circle. Fund. Math. 46 (1958), 187–189.CrossRefGoogle Scholar