In 1890 Poincaré proved the famous Poincaré Recurrence Theorem which in the setting of a separable, metric, finite measure space states that the orbit of almost every point under a measure-preserving transformation eventually returns infinitely close to itself. However, the theorem gives no information on how fast we can expect such points to return. One way to describe the speed of recurrence is to surround a point x in X with a shrinking sequence of balls centred at x and require that the n’th orbit point intersects the n’th target for infinitely many n.
Assuming little more than measure-preserving, Boshernitzan in 1993 gave a lower bound on the radii sequence of the shrinking balls which ensured recurrence almost surely. Until recently, little progress was made on improving Boshernitzans rate, even with much stronger assumptions.
 In this talk I will show that a much more precise result, i.e. a Borel-Cantelli type dichotomy, holds for recurrence for the doubling map. The proof relies on properties of Fourier series and I will discuss how the method can also be used to give a dichotomy for the minimal orbit distance problem.