Benedict Sewell (Rényi Institute):
The α-Kakutani equidistribution problem, and friends.

Fix α ∈ (0,1}, and consider a sequence of partitions of the interval, starting with the trivial one, {[0,1]}, and where the (n+1)st partition is obtained by splitting all maximal intervals in the nth into two, in the ratio α : 1-α.
See the illustration here  for the example of α = ¹/₃.
The question posed to Kakutani: does the set of endpoints of the nth partition become uniformly distributed in the limit?
We will consider the answer to this question, and some of its generalisations/variations: whether its more dimensions, intervals or randomness, there appears to be no limit to the questions one can ask. Naturally, some are open.
Joint work with Mark Pollicott.