Gaps and steps for the sequence nθ mod 1
- 24 October 1967
- journal article
- research article
- Published by Cambridge University Press (CUP) in Mathematical Proceedings of the Cambridge Philosophical Society
- Vol. 63 (4) , 1115-1123
- https://doi.org/10.1017/s0305004100042195
Abstract
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 whichthe step problem: for the setrearranged 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.Keywords
This publication has 4 references indexed in Scilit:
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- On the theory of diophantine approximations. I1 (on a problem of A. Ostrowski)Acta Mathematica Hungarica, 1957
- The distribution of the integers N for which {θN} < φMathematical Proceedings of the Cambridge Philosophical Society, 1950