Uncanny subsequence selections that generate normal numbers

Joseph Vandehey

paper

Uncanny subsequence selections that generate normal numbers

arXiv:1607.03531

Abstract

Given a real number $0. a_{1} a_{2} a_{3} \dots$ 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 $(b - 1)$ .