Dyadic approximation in the middle-third Cantor set
Linfoot Number Theory Seminar
20th May 2020, 4:00 pm – 5:00 pm
Virtual Seminar, CANCELLED
Motivated by a classical question due to Mahler, in 2007 Levesley, Salp, and Velani showed that the Hausdorff measure of the set of points in the middle-third Cantor set which can be approximated by triadic rationals (that is, rationals which have denominators which are powers of 3) at a given rate of approximation satisfies a zero-full dichotomy. More precisely, the Hausdorff measure of the set in question is either zero or full according to, respectively, the convergence or divergence of a certain sum which is dependent on the specified rate of approximation. Naturally, one might also wonder what can be said about dyadic approximation in the middle-third Cantor set. In this talk I will discuss a conjecture on this topic due to Velani, some progress towards this conjecture, and why dyadic approximation is harder than triadic approximation in the middle-third Cantor set. This talk will be based on joint work with Sam Chow (Warwick) and Han Yu (Cambridge).