Distribution Of Sequences Generated By Certain Simply-Constructed Normal Numbers

TL;DR

This paper investigates the distribution properties of sequences derived from certain normal numbers and polynomial functions, revealing conditions under which these sequences are not uniformly distributed.

Contribution

It demonstrates that specific normal numbers produce non-uniformly distributed sequences and extends this to polynomial-generated sequences, providing new insights into distribution behaviors.

Findings

Certain normal numbers yield non-uniform sequences

Polynomial sequences are not uniformly distributed

Results extend previous work by Davenport and Erdős

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.