The minimal denominator function (MDF) quantifies how hard it is to approximate a real number x to a prescribed level of precision through rational numbers. We will explore some quantitative results about the statistics of this function through the lens of homogeneous dynamics.
The first part of the talk will be used to define measure preserving systems, with an emphasis on Lie groups, and translation surfaces.
The second part of the talk will use the setup from the first part to provide quantitative results about the distribution of the MDF. We will prove the existence of a limiting distribution as one of its parameters tends to zero, and generalize it to higher dimensions and statistics of holonomy vectors of translation surfaces. If time allows, we will describe the density function associated to the distribution of the MDF and statistics of saddle connections of translation surface.