Uncanny subsequence selections that generate normal numbers

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 $(b-1)$.