Distribution Of Sequences Generated By Certain Simply-Constructed Normal Numbers

Abstract

In 1949 Wall showed that $x = 0.d_1d_2d_3 \dots$ is normal if and only if $(0.d_nd_{n+1}d_{n+2} \dots)_n$ is a uniformly distributed sequence. In this article, we consider sequences which are slight variants on this. In particular, we show that certain normal numbers of the form $0.a_na_{n+1}a_{n+2} \dots$, where $a_n$ is a sequence of positive integers, give rise in a rather natural way to sequences which are not uniformly distributed. Motivated by a result of Davenport and Erd\H{o}s we also show that for a non-constant integer polynomial the sequence $(0.f(n)f(n+1)f(n+2) \dots)_n$ is not uniformly distributed.