Gaps in the sequence n2ϑ(mod1)

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.