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

• Published in 2021
• Added on 2025-12-18

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 $G_δ$ 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.

Links

• https://arxiv.org/abs/2108.03664v2
• https://arxiv.org/pdf/2108.03664v2

Other information

BibTeX entry

@article{Ontheexistenceofnumberswithmatchingcontinuedfractionanddecimalexpansions,
	key = {Ontheexistenceofnumberswithmatchingcontinuedfractionanddecimalexpansions},
	type = {article},
	title = {On the existence of numbers with matching continued fraction and decimal expansions},
	author = {Pieter Allaart and Stephen Jackson and Taylor Jones and David Lambert},
	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{\_}δ{\$} 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.},
	comment = {},
	date_added = {2025-12-18},
	date_published = {2021-12-26},
	urls = {https://arxiv.org/abs/2108.03664v2,https://arxiv.org/pdf/2108.03664v2},
	collections = {fun-maths-facts},
	url = {https://arxiv.org/abs/2108.03664v2 https://arxiv.org/pdf/2108.03664v2},
	year = 2021,
	urldate = {2025-12-18},
	archivePrefix = {arXiv},
	eprint = {2108.03664},
	primaryClass = {math.NT}
}