Kondo-Saito-Tanaka theorem

Abstract

Given an integer $q\ge 2$, a $q$-normal number is an irrational number $\eta$ such that any preassigned sequence of $k$ digits occurs in the $q$-ary expansion of $\eta$ at the expected frequency, namely $1/q^k$. Given an integer $q\ge 3$, we consider the sequence of primes reduced modulo $q$ and examine various possibilities of constructing normal numbers using this sequence. We create a sequence of independent random variables that mimics the sequence of primes and then show that for almost all outcomes this allows to obtain a normal number.