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"Diophantine approximation: automatic numbers and their generalisations"
Michael Coons
University of Newcastle
In 1844, Liouville gave the first examples of transcendental numbers, providing a criterion for transcendence based on how well a number can be approximated by rationals. This criterion has been refined and generalised, and it may well be the basis for what is now called Diophantine approximation. The first major refinement of Liouville's criterion was made by Thue in 1909, around the time that he was investigating patterns in binary sequences. Thue noted that any binary sequence of length at least four must contain a square. He then asked, is it possible to find an infinite binary sequence that contains no cube, or even no overlap? Thue's questions began an area now called combinatorics on words.

In this presentation, I will discuss the strong relationship between Diophantine approximation and combinatorics on words. This relationship includes the work of Mahler on functional equations, a connection to finite automata, and very recent results on the rational-transcendental dichotomy of associated classes numbers.
Wednesday, 21 May 2014
11:10 a.m. in Math 103
12:00 a.m. Refreshments in Math Lounge 109
Spring 2014 Colloquia & Events
Mathematical Sciences | University of Montana