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

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 $T:={\bigcup }_{b\ge 2}{T}_b$ is nowhere dense and has Hausdorff dimension less than one. Finally, we give several sufficient conditions on bases $b$ and $b^{\prime }$ such that $T_b\cap T_{b^{\prime }}=\varnothing$, and conjecture that this is the case for all $b\ne b^{\prime }$. This question has connections with some deep theorems in Diophantine approximation.

30 pages, new title and a somewhat expanded introduction