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 $D_N \lesssim (\log{ N})^d N^{ -1}$.
Categories: math.CO math.CA math.NT