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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.

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The standard reference for such results is Kuipers and Niederreiter, Uniform Distribution of Sequences.

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