The discrepancy of the Korobov lattice points (Q2901876)

Let \(X=\left(x^{(1)},\dotsc,x^{(k)},\dotsc,x^{(N)}\right)\) be a lattice in the \(s\)-dimensional region NEWLINE\[NEWLINE [0,1)^s=\left\{x=(x_{1},\dotsc,x_{s})\in\mathbb{R}^s\mid 0\leq x_{1}<1,\dotsc,0\leq x_{s}<1\right\}, NEWLINE\]NEWLINE where \(x^{(k)}=\left(x_{1}^{(k)},\dotsc,x_{s}^{(k)}\right)\in [0,1)^s\).NEWLINENEWLINENEWLINEDenote by \( D(X)=\sup\left\{\left.\left|D(X;x,x^\prime\right)\right|, x,x^{\prime}\in[0,1)^{s},x<x^{\prime}\right\} \) the discrepancy of \(X\), where NEWLINE\[NEWLINE D(X;x,x^{\prime})=\displaystyle\frac{1}{N}\sum_{k=1}^N \chi(x^{(k)};x,x^{\prime})-\mathrm{mes}[x,x^{\prime}),NEWLINE\]NEWLINE \(\chi(\cdot;x,x^{\prime})\) is the characteristic function of the half-open region \([x,x^{\prime})=[x_{1},x_{1}^{\prime})\times\dotsb\times [x_{s},x_{s}^{\prime})\) in \([0,1)^s\) and \(\mathrm{mes}[x,x^{\prime})\) is the ordinary measure.NEWLINENEWLINENEWLINELet \(a=(a_{1},\dotsc,a_{s})\) be an arbitrary family of integers in \(\mathbb{Z}^{s}\). The lattices \({\mathcal K}_{N}(a)\) consist of the nodes \(x^{(k)}=\left(\left\{\frac{a_{1}k}{N}\right\},\dotsc ,\left\{\frac{a_{s}k}{N}\right\}\right)\), \(k=1,\dotsc, N,\) where \(\{\alpha\}\) denotes the fractional part of a real number \(\alpha\).NEWLINENEWLINENEWLINEThe behaviour of the quantity \(D({\mathcal K}_{N}(a))\), viewed as a function of \(a\) and \(N\), becomes significantly more complicated when \(s\geq 2\). The bound NEWLINENEWLINE\[NEWLINE D_{N}^{(s)}({\mathcal K})=\displaystyle{\min_{a\in \mathbb{Z}^{s}} D({\mathcal K}_{N}(a))\ll\frac{\log^s N}{N},}\tag{1}NEWLINE\]NEWLINE was proved by Korobov for prime values of \(N\) and by Niederreiter for all positive integers \(N>1\). In this paper, the author proves the following bound for \(s\geq 2\) and \(N\geq 3\), which is stronger than (1): NEWLINE\[NEWLINE D_{N}^{(s)}({\mathcal K})\ll\frac{\log^{s-1}N}{N}\log\log N.NEWLINE\]