On the discrepancy of irrational rotations with isolated large partial quotients: long term effects

Abstract

Setokuchi and Takashima [7] gave general mathematical explanations for the emergence of several parabola-like hills in the behavior of the discrepancies of irrational rotations having single isolated large partial quotient, in somewhat short range of N. We extend estimates in [7] and give some general conditions which ensure repetitions of hills in much longer range of N. For example, in case of 1 − log10 7, our conditions show that more than 2.7 × 1027 repetitions of hills exist, where N runs up to over 6.8 × 1033.