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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 uniformlysmall. A result of Schmidt shows that necessarily $D_N^* \gtrsim (\log{N})N^{-1}$ for infinitely many $N$ and there are several classical constructionsattaining this growth. We describe a type of uniformly distributed sequencethat seems to be completely novel: given $\left\{x_1, \dots, x_{N-1} \right\}$,we construct $x_N$ in a greedy manner $$ x_N = \arg\min_{\min_k |x-x_k| \geqN^{-10}} \sum_{k=1}^{N-1}{1-\log{(2\sin{(\pi |x-x_k|)})}}.$$ 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 impressivemanner. We also establish a discrepancy bound $D_N \lesssim (\log{N})^dN^{-1/2}$ for an analogous construction in higher dimensions and conjecture itto be $D_N \lesssim (\log{N})^d N^{-1}$.

Dynamically Defined Sequences with Small Discrepancy