Well dispersed sequences in \([0,1]^{d}\) (Q1748129)

For a sequence \(\overline{z}=(z_1,z_2,\ldots)\) of points in \([0,1]^d\), define the dispersion measure \(\mu_d(\overline{z})=\inf_{n\geq 1}\inf_{m\geq 1} n^{1/d}\|z_m-z_{m+n}\|\) and \(\alpha_d=\sup_{\overline{z}} \mu_d(\overline{z})\), where \(\|\cdot\|\) denotes the \(L_1\) norm and \(\overline{z}\) ranges over all sequences. In the present paper, the authors show that \(\alpha_d\geq ((2^{d-1}(2^d-1))^{1/d}(1+\sum_{k\geq 1}\frac 1{F_{2k}}))^{-1}\) for \(d\geq 1\), where \(F_n\) denotes the \(n\)-th Fibonacci number defined by \(F_0=0\) and \(F_1=1\). They also obtain upper bounds on \(\alpha_d\), for example \(\alpha_2\leq \frac{2}{1/\sqrt{2}+1/\sqrt{3}}\).