Controlled Divergence of Discrepancy Sums

Abstract

Answering an informal question of K. Park, we show that by fixing some irrational alpha to have a particular standard continued fraction expansion, we may force the associated discrepancy sequences for all x in [0,1), which track the difference between the number of values in the orbit of x under rotation by alpha (modulo one) less than one half versus the number larger than one half, to have maximal values which grow at a prescribed rate.