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- W2971333641 abstract "We describe a curious dynamical system that results in sequences of real numbers in $[0,1]$ with seemingly remarkable properties. Let the function $f:mathbb{T} rightarrow mathbb{R}$ satisfy $hat{f}(k) geq c|k|^{-2}$ and define a sequence via $$ x_n = argmin_x sum_{k=1}^{n-1}{f(x-x_k)}.$$ Such sequences $(x_n)_{n=1}^{infty}$ seem to be astonishingly regularly distributed in various ways (satisfying favorable exponential sum estimates; every interval $J subset [0,1]$ contains $sim |J|n$ elements). We prove $$ W_2left( frac{1}{n} sum_{k=1}^{n}{delta_{x_k}}, dxright) leq frac{c}{sqrt{n}},$$ where $W_2$ is the 2-Wasserstein distance. Much stronger results seem to be true and it seems like an interesting problem to understand this dynamical system better. We obtain optimal results in dimension $d geq 3$: using $G(x,y)$ to denote the Green's function of the Laplacian on a compact manifold, we show that $$ x_n = argmin_{x in M} sum_{k=1}^{n-1}{G(x,x_k)} quad mbox{satisfies} quad W_2left( frac{1}{n} sum_{k=1}^{n}{delta_{x_k}}, dxright) lesssim frac{1}{n^{1/d}}.$$" @default.
- W2971333641 created "2019-09-05" @default.
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- W2971333641 date "2019-08-29" @default.
- W2971333641 modified "2023-09-27" @default.
- W2971333641 title "Positive-definite Functions, Exponential Sums and the Greedy Algorithm: a curious Phenomenon" @default.
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