For every unit vector $\sigma\in\Sigma_{ d-1}$ and every $r\ge0$, let % % \begin{ displaymath} P_{ \sigma,r}=[-1,1]^d\cap\{ t\in\Rr^d:t\cdot\sigma\le r\} \end{ displaymath} % % denote the intersection of the cube $[-1,1]^d$ with a half-space containing the origin $0\in\Rr^d$. We prove that if $N$ is the $d$-th power of an odd integer, then there exists a distribution $\PPP$ of $N$ points in $[-1,1]^d$ such that % % \begin{ displaymath} \sup_{ r\ge0} \int_{ \Sigma_{ d-1}}\vert\card(\PPP\cap P_{ \sigma,r})-N2^{ -d} \vert P_{ \sigma,r}\vert\vert\,\dd\sigma \le c_d(\log N)^d, \end{ displaymath} % % generalizing an earlier result of Beck and the first author.
Categories: math.NT