Dynamical Systems Seminar

We discuss some topics about number theory including continued fractions, Hausdorff measure, p- adic analysis and analytic number theory in the preliminary knowledge part. In the next section, we investigate the problem of how well points in finite dimensional \(p\)-adic solenoids can be approximated by rationals. The setting we work in was previously studied by Palmer, who proved analogues of Dirichlet’s theorem and the Duffin-Schaeffer theorem. We prove a complementary result, showing that the set of badly approximable points has maximum Hausdorff dimension. Our proof is a simple application of the elegant machinery of Schmidt’s game. Moreover, we compute the probability mass function of the random variable which returns the smallest denominator of a reduced fraction in a randomly chosen real interval of radius \(\delta\). As an application, we prove that the expected value of the smallest denominator is asymptotic, as \(\delta\to 0\), to \((8 \sqrt{2}/π^2)\delta^{−1/2}\).