Submission date: 9 July 2017

Abstract:

We give a construction of an absolutely normal real number x such that for every integer b greater than or equal to 2, the discrepancy of the first N terms of the sequence (b^n x mod 1)_{n ≥ 0} is of asymptotic order O(N^{-1/2}). This is below the order of discrepancy which holds for almost all real numbers. Even the existence of absolutely normal numbers having a discrepancy of such a small asymptotic order was not known before.

Mathematics Subject Classification: Primary 11K16, Secondary 11-Y16, 68-04

Keywords and phrases:

Full text arXiv 1707.02628: pdf, ps.