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

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.