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Dynamically Defined Sequences with Small Discrepancy (1902.03269v2)

Published 8 Feb 2019 in math.CO, math.CA, and math.NT

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 $D_N \lesssim (\log{N})d N{-1}$.

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