On the existence of numbers with matching continued fraction and decimal expansions

TL;DR

This paper investigates numbers called Trott numbers, which have matching continued fraction and base-$b$ expansions, characterizing bases where they exist and analyzing their topological and measure-theoretic properties.

Contribution

It characterizes the bases allowing Trott numbers, shows the structure of their sets, and explores intersections between different bases, connecting to Diophantine approximation.

Findings

Trott numbers exist only for certain bases.

The set of Trott numbers in these bases is a complete $G_\delta$ set.

The union of Trott numbers across all bases is nowhere dense with Hausdorff dimension less than one.

Abstract

A Trott number is a number $x\in(0,1)$ whose continued fraction expansion is equal to its base $b$ expansion for a given base $b$, in the following sense: If $x=[0;a_1,a_2,\dots]$, then $x=(0.\hat{a}_1\hat{a}_2\dots)_b$, where $\hat{a}_i$ is the string of digits resulting from writing $a_i$ in base $b$. In this paper we characterize the set of bases for which Trott numbers exist, and show that for these bases, the set $T_b$ of Trott numbers is a complete $G_\delta$ set. We prove moreover that the union $T:=\bigcup_{b\geq 2} T_b$ is nowhere dense and has Hausdorff dimension less than one. Finally, we give several sufficient conditions on bases $b$ and $b'$ such that $T_b\cap T_{b'}=\emptyset$, and conjecture that this is the case for all $b\neq b'$. This question has connections with some deep theorems in Diophantine approximation.