On the existence of numbers with matching continued fraction and base b 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 $b'$ such that $T_b\cap T_{b'}=\emptyset$, and conjecture that this is the case for all $b\ne b'$. This question has connections with some deep theorems in Diophantine approximation.