2019/08/29 by Brown, Louis, Steinerberger, Stefan
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1908.11228
We describe a curious dynamical system that results in sequences of real numbers in [0,1] with seemingly remarkable properties. Let the function f:\mathbbT → ℝ satisfy f(k) ≥ c|k|-2 and define a sequence via xn = argminx ∑k=1n-1f(x-xk). Such sequences (xn)n=1∞ seem to be astonishingly regularly distributed in various ways (satisfying favorable exponential sum estimates; every interval J ⊂ [0,1] contains ∼ |J|n elements). We prove W2( (1)/(n) ∑k=1nδxk, dx) ≤ (c)/(√(n)), where W2 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 ≥ 3: using G(x,y) to denote the Green's function of the Laplacian on a compact manifold, we show that xn = argminx ∈ M ∑k=1n-1G(x,xk) satisfies W2( (1)/(n) ∑k=1nδxk, dx) \lesssim \frac1n1/d.