Diophantine Approximation, Games, and Fractals

By Asaf Reich

October 2, 2012

Abstract

In diophantine approximation, we want to make more precise and quantitative the idea that any real number has rationals getting arbitrarily close to it, by asking how fast the distance to them can go to zero as a function of their denominators. Surprisingly, it turns out that a useful technique for proving some results in this topic is that of 'topological games', abstract games in which players' moves are choices of subsets in a space. Also surprisingly, when we ask about approximating the reals in a given subset of R, it turns out that some nice generalizations hold if the subset in question is fractal-like!