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International Journal of Mathematics and Mathematical Sciences
Volume 10, Issue 1, Pages 131-134
http://dx.doi.org/10.1155/S0161171287000164

Gaps in the sequence n2ϑ(mod1)

Department of Mathematics, Santa Clara University, Santa Clara 95053, CA, USA

Received 16 September 1985

Copyright © 1987 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let ϑ be an irrational number and let {t} denote the fractional part of t. For each N let I0,I1,,IN be the intervals resulting from the partition of [0,1] by the points {k2ϑ}, k=1,2,,N. Let T(N) be the number of distinct lengths these intervals can assume. It is shown that T(N). This is in contrast to the case of the sequence {nϑ}, where T(N)3.