Insertion in constructed normal numbers

TL;DR

This paper investigates the process of inserting digits into constructed base-$b$ normal numbers to achieve normality in base $(b+1)$, addressing a fundamental question in number theory and normal number construction.

Contribution

It introduces methods for inserting digits into base-$b$ normal numbers to produce normal numbers in base $(b+1)$, advancing understanding of normal number transformations.

Findings

Demonstrates conditions under which insertion preserves normality

Provides constructions for base transition normal numbers

Advances theoretical understanding of normal number manipulations

Abstract

Defined by Borel, a real number is normal to an integer base $b$, greater than or equal to $2$, if in its base-$b$ expansion every block of digits occurs with the same limiting frequency as every other block of the same length. We consider the problem of insertion in constructed base-$b$ normal expansions to obtain normality to base $(b+1)$.