From Randomness in Two Symbols to Randomness in Three Symbols

Abstract

In 1909 Borel defined normality as a notion of randomness of the digits of the representation of a real number over certain base (fractional expansion). If we think of the representation of a number over a base as an infinite sequence of symbols from a finite alphabet A, we can define normality directly for words of symbols of A: A word x is normal to the alphabet A if every finite block of symbols from A appears with the same asymptotic frequency in x as every other block of the same length. Many examples of normal words have been found since its definition, being Champernowne in 1933 the first to show an explicit and simple instance. Moreover, it has been characterized how we can select subsequences of a normal word x preserving its normality, always leaving the alphabet A fixed. In this work we consider the dual problem which consists of inserting symbols in infinitely many positions of a given word, in such a way that normality is preserved. Specifically, given a symbol b that is not present in the original alphabet A and given a word x that is normal to the alphabet A we solve how to insert the symbol b in infinitely many positions of the word x such that the resulting word is normal to the expanded alphabet A ∪{b}.