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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
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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.

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

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References

REFERENCES

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