Gaps in the sequence \(n^ 2\vartheta \,(mod\,1)\) (Q1090701)

Two ingredients are put together in this paper: some simple lemmas on fractional parts of \{\(n\vartheta\}\) and an upper bound for the number of divisors of a given number. The result is the following theorem: Let \(\vartheta\) be irrational. Consider the sequence \(\{k^ 2\vartheta \}\), \(1\leq k\leq N\). Then a partition of [0,1] into \(N+1\) subintervals results. Let T(N) be the number of distinct lengths which show up in this set of intervals. Then \(T(N)\gg N^{1-\delta}\). [The paper actually gives a sharper bound and states a generalization to the sequence \(\{k^ p\vartheta \}\), \(p\geq 2.]\)