From randomness in two symbols to randomness in three symbols

TL;DR

This paper explores how to insert new symbols into infinite sequences to preserve their normality, extending the concept from fixed alphabets to expanded ones with additional symbols.

Contribution

It introduces methods for inserting symbols into normal words to maintain normality over larger alphabets, addressing a dual problem to subsequence selection.

Findings

Established procedures for symbol insertion preserving normality

Extended normality concepts to expanded alphabets with new symbols

Provided explicit constructions for normal words over larger alphabets

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 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 infinite positions of a given word, in such a way that normality is preserved. Specifically, given a symbol $s$ that is not present on the original alphabet $A$ and given a word $x$ that is normal to the alphabet $A$ we solve how to insert the symbol $s$ in infinite positions of the word $x$ such that the resulting word is normal to the expanded alphabet $A\cup \{s\}$.