BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Taylor Jones (University of North Texas) DTSTART:20211005T130000Z DTEND:20211005T133000Z DTSTAMP:20260423T021325Z UID:OWNS/60 DESCRIPTION:Title: On the Existence of Numbers with Matching Continued Fraction and Decimal Exp ansion\nby Taylor Jones (University of North Texas) as part of One Wor ld Numeration seminar\n\n\nAbstract\nA Trott number in base 10 is one whos e continued fraction expansion agrees with its base 10 expansion in the se nse that $[0\;a_1\,a_2\,\\dots] = 0.(a_1)(a_2) \\cdots$ where $(a_i)$ repr esents the string of digits of $a_i$. As an example $[0\;3\,29\,54\,7\,\\d ots] = 0.329547\\cdots$.\nAn 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 wh ich $b$? We discuss the history of the problem\, and give a heuristic proc ess for constructing such numbers. We show that $T_{10}$ is indeed non-emp ty\, and uncountable. With more delicate techniques\, a complete classific ation may be given to all $b$ for which $T_b$ is non-empty. We also discus s some further results\, such as a (non-trivial) upper bound on the Hausdo rff dimension of $T_b$\, as well as the question of whether the intersecti on of $T_b$ and $T_c$ can be non-empty.\n LOCATION:https://researchseminars.org/talk/OWNS/60/ END:VEVENT END:VCALENDAR