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Given a positive increasing function $\varphi$, we show that for a fullmeasure set of vectors $\alpha\in \mathbb{R}^d$, the maximal ergodicdiscrepancy of the $d$-linear form sequence $\left\{\sum_{1\le i\le d}k_i\alpha_i \mod 1\right\}_{1\le k_i\le N\atop 1\le i\le d}$ relative tointervals in $[0,1)$ has an absolute upper bound of $C(\alpha,\varphi)\cdot(\log N)^d \varphi^{\max\{d,3\}}(\log \log N)$ if $\sum_{n=1}^{\infty}\frac{1}{\varphi(n)}$ converges.

Absolute Bounds for Ergodic Deviations of Linear Form Sequences Relative to Intervals in $\mathbb{T}^1$