Dynamically defined sequences with small discrepancy

We study the problem of constructing sequences (xn)n=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_n)_{n=1}^{\infty }$$\end{document} on [0, 1] in such a way that DN∗=sup0≤x≤1#1≤i≤N:xi≤xN-x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_N^* = \sup _{0 \le x \le 1} \left| \frac{ \#\left\{ 1 \le i \le N: x_i \le x \right\} }{N} - x \right| \end{aligned}$$\end{document}is small. A result of Schmidt shows that for all sequences sequences (xn)n=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_n)_{n=1}^{\infty }$$\end{document} on [0, 1] we have DN∗≳(logN)N-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_N^* \gtrsim (\log {N}) N^{-1}$$\end{document} for infinitely many N, several classical constructions attain this growth. We describe a type of uniformly distributed sequence that seems to be completely novel: given x1,⋯,xN-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ x_1, \dots , x_{N-1} \right\} $$\end{document}, we construct xN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_N$$\end{document} in a greedy manner xN=argminmink|x-xk|≥N-10∑k=1N-11-log(2sin(π|x-xk|)).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} x_N = \arg \min _{\min _k |x-x_k| \ge N^{-10}} \sum _{k=1}^{N-1}{1-\log {(2\sin {(\pi |x-x_k|)})}}. \end{aligned}$$\end{document}We prove that DN∗≲(logN)N-1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_N^* \lesssim (\log {N}) N^{-1/2}$$\end{document} and conjecture that DN∗≲(logN)N-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_N^* \lesssim (\log {N}) N^{-1}$$\end{document}. Numerical examples illustrate this conjecture in a very impressive manner. We also establish a discrepancy bound DN∗≲(logN)dN-1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_N^* \lesssim (\log {N})^d N^{-1/2}$$\end{document} for an analogous construction in higher dimensions and conjecture it to be DN∗≲(logN)dN-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_N^* \lesssim (\log {N})^d N^{-1}$$\end{document}.