On the existence of Trott numbers relative to multiple bases

Abstract: Trott numbers are real numbers in the interval $(0,1)$ whose continued fraction expansion equals their base-$b$ expansion, in a certain liberal but natural sense. They exist in some bases, but not in all. In a previous OWNS talk, T. Jones sketched a proof of the existence of Trott numbers in base 10. In this talk I will discuss some further properties of these Trott numbers, and focus on the question: Can a number ever be Trott in more than one base at once? While the answer is almost certainly "no", a full proof of this seems currently out of reach. But we obtain some interesting partial answers by using a deep theorem from Diophantine approximation.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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One World Numeration seminar

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