Mathematics Colloquia and Seminars

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The Markoff numbers are a fascinating sequence of positive integers which appear in surprising ways across different branches of mathematics. The Markoff numbers were introduced in 1880 as the solutions to a certain Diophantine equation and were used to classify the worst-approximable irrationals. In the 1970s, the connection between the Markoff numbers and the lengths of simple closed geodesics on punctured torus brought hyperbolic geometry into the picture. More recently, the number theoretic properties of the Markoff numbers themselves have been of great interest. In this talk, we will discuss properties of the Markoff sequence itself. Specifically, we will address the following question: Does the sequence of Markoff numbers satisfy Benford's law for first digits? A sequence satisfies Benford's law if the probability of observing a first digit equal to d converges to $P(d)=\log_{10}(1+1/d)$. Benford's law is closely connected to the theory of uniform distribution modulo 1, which we will also discuss.