Hello! Ready to read the full proof? You can download the notes below.
Scroll further down for a couple of the pretty pictures we’ve been making and a brief explanation of them.
To all my wonderful collaborators - thank you! I have learned so much from you and it is a great pleasure working with you all.
Poster Notes - Birkhoff Summation of Irrational Rotations.pdf
I expanded these notes into my honors thesis, which adds some explanatory background.
My poster design is based on the excellent video below. Thanks to Mike Morrison for the idea and the template, and to Mary, my technical writing professor, for sharing the video with me.
https://youtu.be/1RwJbhkCA58?si=ho4Stsu1HcWeq_sO
There’s also a second generation video which is excellent as well!
Here’s the template I used for my poster, by Mike Morrison and Raphael Bailo:
https://www.overleaf.com/latex/templates/better-poster-latex-template/gmkgjvxqbyyt
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This is the ‘range graph’ for the Birkhoff sum $S_N$ of the golden mean as the number of terms in the sum increases. Beautiful! Don’t you think?
Here’s the definition of the sum, where { } indicate the fractional part.
$$ S_N=S(\rho,N,x)=\sum_{i=0}^{N-1}\left(\{x+i\rho\}-\frac{1}{2}\right) $$
The range graph for a given sum $S_N$ shows the length of the interval between the minimum and maximum values of $S_N(x)$ across all values of $x\in[0,1)$. When $N$ is a continued fraction denominator, this range is quite small (and in fact approaches a known value as discussed in the full notes). In the graph above, this can be seen in the dramatic dips that occur when $N$ is a Fibonacci number.
Here’s another pretty one using the fractional part of $e$ as our rotation number:
