To my knowledge, the best upper bound for the discrepancy of sequences of the type $(n\alpha) (\mod 1), n=1,2,...$ is $\frac{ND_N(\alpha)}{\log N\log\log N}\to \frac{2}{\pi^2}$ in measure.
My first question is this: Given a particular double sequence $a_{ij}=i\alpha+j\beta (\mod 1)$, what is the optimal bound on the discrepancy in terms of $\alpha$ and $\beta$? For $1\le i,j\le N$ it is straight forward to show that the sequence of $N^2$ terms has a discrepancy no larger than that of its "best direction", i.e. $D_{N^2}(a_{ij})\le \min\{D_N(i\alpha),D_N(j\beta)\}.$ I would very much like to know if this could be improved.
Secondly, I'm interested in the discrepancy of polynomial sequences. Given a polynomial of order $d$, what is the discrepancy of $(p(n)) (\mod 1)$, $n=1,2,...$, depending on $d$ and the coefficients?
I would be very grateful for any enlightenment or references you could give.