Birkhoff Sums Over the Golden Rotation

Update posted November 2016: Some notes [PDF] and Slides [PDF] to a talk given at BU in February 2015.
In this project with Folkert Tangerman from Stony Brook we look at the Birkhoff sum

with g(x) = 2 log(2 sin(pi x)), where alpha = (sqrt(5)-1)/2 is the golden mean. In a project of 2008 we showed that S_n/log(n)² is bounded. In the current work, we relate the boundedness of S_n/log(n) to a limiting function f(x) obtained as a limiting difference between consecutive continued fraction expansions. While the function s_n(x) = S_{[x q_n](alpha)/log(q_n)} only converges weakly, the sequence of functions

f_n(x) = S_{[x q_n]}(p_n+1/q_n+1) - S_{[x q_n]}(p_n/q_n)

converges pointwise to a function f(x) on [0,1].

This project started during spring break 2010. Folkert visited Cambridge once in April. Otherwise we worked electronically or by phone. The paper got finalized during a meeting in New London (MA) on May 24, 2010.