On the lack of equidistribution on fat Cantor sets

Given an irrational rotation, it is straightforward to see that for every Cantor set C, there is a dense set of points whose orbit doesn't intersect C. On the other hand, if C is a fat Cantor set (that is, of positive Lebesgue measure), almost every point visits C with a frequency equal to the measure of C. But what other frequencies of visits to C may occur? In the words of a recent MathOverflow post [1], what is the Birkhoff spectrum of fat Cantor sets?

We give a first answer to this question by showing that every irrational rotation allows for certain fat Cantor sets C whose Birkhoff spectrum is maximal, that is, equal to the interval [0,Leb(C)]. In this talk, I will focus on discussing some of the basic tools behind this result and extensions of it.

[1] D. Kwietniak, Possible Birkhoff spectra for irrational rotations, MathOverflow (2020), https://mathoverflow.net/q/355860 (version: 2020-03-27).