This week we'll be joined by Zhuo Zhang! Zhuo is an undergraduate who recently collaborated on research with their colleagues Shreyas Singh and Alex Jin under Prof. AJ Hildebrand. They'll be here to discuss this project, which they also presented recently at this year’s Young Mathematicians Conference at Ohio State University:
Continued Fraction and the Gauss-Kuzmin Distribution
It is well-known that any irrational number x has a unique representation as an infinite continued fraction. Gauss and Kuzmin showed that, for almost all real numbers x, the frequency of the digit k (positive integer) in the continued fraction expansion of x is given by P(k) = log2(1 + 1/k(k+2)) (I will talk more about this in the presentation). In our work we generalized this result in several directions. First, we considered the frequency with which the continued fraction digits of a random real number x belong to a given subset A ⊆ N. Second, we considered the frequencies of finite strings of digits in the continued fraction expansion of a random real number. Third, we proved a unique characterization of the Gauss Kuzmin distribution in terms of the relative frequencies of strings and their reversals in the continued fraction expansion of random real numbers.
As usual, there will be free pizza courtesy of the IGL!
