On the existence of numbers with matching continued fraction and base b expansions

Trott number is a number x∈ (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,… ] , then x=(0.â_1â_2… )_b , where â_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_δ set. We prove moreover that the union T:=⋃ _b≥ 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∩ T_b'=∅ , and conjecture that this is the case for all b b' . This question has connections with some deep theorems in Diophantine approximation.