On the Existence of Numbers with Matching Continued Fraction and Decimal Expansion

Abstract: A Trott number in base 10 is one whose continued fraction expansion agrees with its base 10 expansion in the sense that $[0;a_1,a_2,\dots] = 0.(a_1)(a_2) \cdots$ where $(a_i)$ represents the string of digits of $a_i$. As an example $[0;3,29,54,7,\dots] = 0.329547\cdots$. An analogous definition may be given for a Trott number in any integer base $b>1$, the set of which we denote by $T_b$. The first natural question is whether $T_b$ is empty, and if not, for which $b$? We discuss the history of the problem, and give a heuristic process for constructing such numbers. We show that $T_{10}$ is indeed non-empty, and uncountable. With more delicate techniques, a complete classification may be given to all $b$ for which $T_b$ is non-empty. We also discuss some further results, such as a (non-trivial) upper bound on the Hausdorff dimension of $T_b$, as well as the question of whether the intersection of $T_b$ and $T_c$ can be non-empty.

dynamical systemsnumber theory

Audience: researchers in the topic

( slides )

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One World Numeration seminar

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