Distribution Of Sequences Generated By Certain Simply-Constructed Normal Numbers

2015/11/04 by Demi Allen, Allen, Demi, Sky Brewer +1
Computer Science · Mathematics · #11J71 #Analytic Number Theory Research #Benford’s Law and Fraud Detection #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Primary 11K06 #Secondary 11K16 #math.NT #msc:11J71 #msc:11K06 #msc:11K16

paper · pdf · doi:10.48550/arxiv.1511.01789

arxiv created 2015/11/04 · openalex publication_date 2015/11/04 · arxiv updated 2015/11/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In 1949 Wall showed that x = 0.d₁d₂d₃ … is normal if and only if (0.d_nd_n+1d_n+2 …)_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+1a_n+2 …, 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 Erdős we also show that for a non-constant integer polynomial the sequence (0.f(n)f(n+1)f(n+2) …)_n is not uniformly distributed.

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