Wednesday, October 4
Milner 317, 12:30 PM
Title: Martingales and continued fractions
Abstract: We will
discuss a complete system of martingale differences which encodes
information about the continued fraction expansion of real numbers.
Using results of R. F. Gundy we will prove that every measurable
function on R/Z which is finite almost everywhere has an almost
everywhere convergent series representation in terms of continued
fractions. Also of interest are problems related to the
Duffin-Schaeffer conjecture. We will show how maximal inequalities for
partial sums of martingale differences can be used to give non-trivial
upper bounds for the variance of a class of weighted sums of
Duffin-Schaeffer type.