Uncanny subsequence selections that generate normal numbers

TL;DR

This paper explores special subsequences of normal numbers, demonstrating new constructions that preserve normality and showing that removing certain digits from a normal number yields another normal number in a different base.

Contribution

It introduces novel methods for selecting subsequences of normal numbers and proves that digit removal can produce normal numbers in lower bases.

Findings

Arithmetic progression subsequences preserve normality.

Recursively defined subsequences can produce normal numbers.

Removing all (b-1) digits from a normal number yields a normal number in base (b-1).

Abstract

Given a real number $0.a_1a_2 a_3\dots$ that is normal to base $b$, we examine increasing sequences $n_i$ so that the number $0.a_{n_1}a_{n_2}a_{n_3}\dots$ are normal to base $b$. Classically it is known that if the $n_i$ form an arithmetic progression then this will work. We give several more constructions, including $n_i$ that are recursively defined based on the digits $a_i$. Of particular interest, we show that if a number is normal to base $b$, then removing all the digits from its expansion which equal $(b-1)$ leaves a base-$(b-1)$ expansion that is normal to base $(b-1)$.