Absolute Bounds for Ergodic Deviations of Toral Translations Relative to Triangles in $\mathbb{T}^2$

TL;DR

This paper establishes bounds on the ergodic discrepancy of toral translations relative to triangles in , extending Beck's work by including additional factors to account for small divisors, with results depending on the convergence of a series involving .

Contribution

It provides new upper and lower bounds for ergodic discrepancies relative to triangles in , incorporating a  factor to handle small divisors, extending previous results for boxes.

Findings

Bounded discrepancy when series converges

Unbounded discrepancy when series diverges

Additional  factor necessary for small divisor control

Abstract

Following Beck's work on toral translations relative to straight boxes in $\mathbb{T}^n$, we prove a weaker upper bound and the same lower bound for ergodic discrepancies of toral translations relative to a triangle in $\mathbb{T}^2$. Specifically, given a positive increasing function $\varphi(n)$, we show that for a full measure set of translation vectors $\alpha\in \mathbb{T}^2$, if the series $\sum_{N=1}^{\infty} \frac{1}{\varphi (N)} $ converges, then the maximal discrepancy of toral translations relative to the triangles of a given slope $\tau$ is bounded from above by $Const(\alpha,\tau) (\log N)^2 \varphi^2(\log \log N)$, and there would be infinitely many $N$'s such that the maximal discrepancy is greater than $(\log N)^2\varphi(\log \log N)$ if the series $\sum_{N=1}^{\infty} \frac{1}{\varphi (N)} $ diverges. An important difference between our result and that of Beck' is an additional factor $\varphi(\log \log N)$, which is necessary in our proof for controlling the new small divisors created by the hypotenuse.