DSNT Seminar: Uniform Diophantine approximation via continued fractions

Abstract: In one-dimensional Diophantine approximation, by using the theory of continued fractions, Khintchine’s theorem and Jarnik’s theorem are concerned with the size of the set of real numbers for which the partial quotients in their continued fraction expansions grows with a certain rate. Whereas it was observed that the improvability of Dirichlet’s theorem is concerned with the growth of the product of consecutive partial quotients in the continued fraction expansion of a real number. In this talk, I will describe some metrical properties of the product of an arbitrary block of consecutive partial quotients raised to different powers in continued fractions, including the Lebesgue measure-theoretic result and the Hausdorff dimensional result.

This is a seminar in our seminar series on Dynamical Systems and Number Theory (DSNT).