Selfsimilarity in the Birkhoff sum of the cotangent function

TL;DR

This paper proves that the Birkhoff sum of the cotangent function with golden ratio rotation converges to a self-similar function, enabling precise calculations of large sums and revealing complex limiting behaviors.

Contribution

It introduces a new self-similar limiting function for the Birkhoff sum of cotangent with golden ratio rotation, extending understanding beyond classical ergodic results.

Findings

The Birkhoff sum converges to a self-similar function s(x) on [0,1].

The limiting function s(x) can be computed analytically.

This approach allows accurate calculation of extremely large sums like S(10^100)/10^100.

Abstract

We prove that the Birkhoff sum S(n)/n = (1/n) sum_(k=1)^(n-1) g(k A) with g(x) = cot(Pi x) and golden ratio A converges in the sense that the sequence of functions s(x) = S([ x q(2n)])/q(2n) with Fibonacci numbers q(n) converges to a self similar limiting function s(x) on [0,1]. The function s(x) can be computed analytically. This allows to determine values like S(10^100)/10^100 accurately without that it ever would be possible to add up so many terms for this random walk. The random variables added up are Cauchy distributed random variables with almost periodic correlation. While for any continuous function g, the Birkhoff limiting function is s(x)=M x by Birkhoff's ergodic theorem, we get so examples of random variables X(n), where the limiting function of S([x n])/n converges to a nontrivial selfsimilar function s(x) along subsequences for one initial point. Hardy and Littlewood have studied the Birkhoff sum for g'(x)= -Pi csc^2(Pi x) and shown that the corresponding Birkhoff sum S'(n)/n^2 stays bounded. Sinai and Ulcigrai have found a limiting distribution for g(x) if both the rotation number A and the initial point t are integrated over. We fix the golden ratio A, start with fixed t=0 and show that the rescaled random walk converges along subsequences. The analysis sheds light on the pictures seen in previous papers with Lesieutre and Tangerman on the antiderivative G(x).