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A Trott number is a number $x\in(0,1)$ whose continued fraction expansion isequal 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 $T:=\bigcup_{b\geq 2} T_b$ isnowhere dense and has Hausdorff dimension less than one. Finally, we giveseveral sufficient conditions on bases $b$ and $b'$ such that $T_b\capT_{b'}=\emptyset$, and conjecture that this is the case for all $b\neq b'$.This question has connections with some deep theorems in Diophantineapproximation.

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