Unique Representations of Real Numbers in Non-Integer Bases

Liam Orovec, Department of Pure Mathematics, University of Waterloo

When looking at the representation of numbers in non-integer bases, $\beta$-expansions, we often find an infinite number of expansions for any given real number under any given base. We look at finding, given a fixed positive real number $x$, the smallest base $q_s(x)$ for which $x$ has a unique $q_s(x)$-expansion. Beginning with $x=1$ we find the ever present Thue-Morse sequence will be helpful throughout the talk. Having found our constant $q_{KL}=q_s(1)$, the Komornik-Loreti constant, we will explore when $q_s(x)<q_{KL}$. The majority of this talk will follow the Results due to Derong Kong which covers the case where are expansions have only digits 0 and 1, in what time that remains we will look at generalizing these results for larger alphabets.

MC 5479