Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dynamically Defined Sequences with Small Discrepancy   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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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: physics
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-th
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom