  Given a positive increasing function $\varphi$, we show that for a full
measure set of vectors $\alpha\in \mathbb{R}^d$, the maximal ergodic
discrepancy 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 to
intervals 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.
