Number Theory Seminar

Tuesday, March 7, 2023 11:30 am - 11:30 am EST

Liam Orovec, Department of Pure Mathematics, University of Waterloo

"Unique Representations of Real Numbers in Non-Integer Bases"

When looking at the representation of numbers in non-integer bases, β-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 qs(x) for which x has a unique qs(x)-expansion.
Beginning with x = 1 we find the ever present Thue-Morse sequence will be helpful throughout the
talk. Having found our constant qKL = qs(1), the Komornik-Loreti constant, we will explore when
qs(x) < qKL. 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