On strong irregularities of the distribution of {nα} sequences

Abstract

Let U={u _n be a sequence in [0, 1]^k and $\Delta _N^U = \mathop {\sup }\limits_l |\Delta _N (I)| = \mathop {\sup }\limits_l \left| {\sum\limits_{\mathop {u_n \in l}\limits_{1 \leqq n \leqq N} } {1 - N|l|} } \right|$ (l a subinterval of [0, 1]^k). By Schmidt’s theorem Δ _N > c ₁ log N for any N if k = 2 while for k = 1 only $\overline {\lim } \frac{{\Delta _N }} {{\log N}} > c_2 > 0$ holds and we have sequences (e.g. {n α} sequences) for which Δ _N ≦1 for infinitely many N. Inspite of this fact we have the following Theorem: Let u _n = {nα}. With a suitable δ∈(0, 1) and for every N>N ₀

$$\Delta _n > c_3 \log N$$

holds for all but at most N ^δ values of n, 1≦n≦N. (Here c ₃>0 is an absolute constant.