On the construction of absolutely normal numbers

2017/07/09 by Aistleitner, Christoph, Becher, Verónica, Scheerer, Adrian-Maria +1 · 1 citation
#68-04 #FOS: Mathematics #Number Theory (math.NT) #Primary 11K16 #Secondary 11-Y16

paper · doi:10.48550/arxiv.1707.02628

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 (bⁿ 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.

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