# Digit expansions of numbers in different bases

## TL;DR

This paper advances understanding of a folklore number theory conjecture about integers with binary digits in multiple bases, explores the density of such numbers, and generalizes to a measure-zero set in the unit interval.

## Contribution

It provides the first progress on the conjecture, analyzes the density transition, and generalizes to sets with zero Hausdorff dimension for missing digits in all bases ≥ 3.

## Key findings

- Confirmed the conjecture for specific cases
- Discovered a phase transition in the density of binary-digit numbers
- Proved the set of numbers missing certain digits in all bases ≥ 3 has zero Hausdorff dimension

## Abstract

A folklore conjecture in number theory states that the only integers whose expansions in base $3,4$ and $5$ contain solely binary digits are $0, 1$ and $82000$. In this paper, we present the first progress on this conjecture. Furthermore, we investigate the density of the integers containing only binary digits in their base $3$ or $4$ expansion, whereon an exciting transition in behaviour is observed. Our methods shed light on the reasons for this, and relate to several well-known questions, such as Graham's problem and a related conjecture of Pomerance. Finally, we generalise this setting and prove that the set of numbers in $[0, 1]$ who do not contain some digit in their $b$-expansion for all $b \geq 3$ has zero Hausdorff dimension.