On the smallest base in which a number has a unique expansion

Abstract: For $x>0$, let $U(x)$ denote the set of bases $q \in (1,2]$ such that $x$ has a unique expansion in base $q$ over the alphabet $\{0,1\}$, and let $f(x)=\inf U(x)$. I will explain that the function $f(x)$ has a very complicated structure: it is highly discontinuous and has infinitely many infinite level sets. I will describe an algorithm for numerically computing $f(x)$ that often gives the exact value in just a small finite number of steps. The Komornik-Loreti constant, which is $f(1)$, will play a central role in this talk. This is joint work with Derong Kong, and builds on previous work by Kong (Acta Math. Hungar. 150(1):194--208, 2016).

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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One World Numeration seminar

Series comments: Description: Online seminar on numeration systems and related topics

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