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# Texas A&M Number Theory Seminar

##
Department of Mathematics

Milner 317

Wednesdays, 12:30-1:30 PM

Alan Haynes

University of Texas at Austin

**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.