Dynamically Defined Sequences with Small Discrepancy

Abstract

We study the problem of constructing sequences $(x_n)_{n=1}^{\infty}$ on $[0,1]$ in such a way that $$ D_N^* = \sup_{0 \leq x \leq 1} \left| \frac{ \left\{1 \leq i \leq N: x_i \leq x \right\}}{N} - x \right|$$ is uniformly small. A result of Schmidt shows that necessarily $D_N^* \gtrsim (\log{N}) N^{-1}$ for infinitely many $N$ and there are several classical constructions attaining this growth. We describe a type of uniformly distributed sequence that seems to be completely novel: given $\left\{x_1, \dots, x_{N-1} \right\}$, we construct $x_N$ in a greedy manner $$ x_N = \arg\min_{\min_k |x-x_k| \geq N^{-10}} \sum_{k=1}^{N-1}{1-\log{(2\sin{(\pi |x-x_k|)})}}.$$ We prove that $D_N \lesssim (\log{N}) N^{-1/2}$ and conjecture that $D_N \lesssim (\log{N}) N^{-1}$. Numerical examples illustrate this conjecture in a very impressive manner. We also establish a discrepancy bound $D_N \lesssim (\log{N})^d N^{-1/2}$ for an analogous construction in higher dimensions and conjecture it to be $D_N \lesssim (\log{N})^d N^{-1}$.