Greedy energy minimization can count in binary: point charges and the van der Corput sequence

Abstract

This paper establishes a connection between a problem in Potential Theory and Mathematical Physics, arranging points so as to minimize an energy functional, and a problem in Combinatorics and Number Theory, constructing 'well-distributed' sequences of points on $[0,1)$. Let $f:[0,1]\to \mathbb{R}$ be (i) symmetric $f(x)=f(1-x)$, (ii) twice differentiable on $(0,1)$, and (iii) such that $f^{\prime \prime }(x)>0$ for all $x\in (0,1)$. We study the greedy dynamical system, where, given an initial set $\{x_0,\dots ,x_{N-1}\}\subset [0,1)$, the point $x_N$ is obtained as $$x_N=\arg \underset{x\in [0,1)}{\min }\sum _{k=0}^{N-1}f(|x-x_k|).$$ We prove that if we start this construction with the single element $x_0=0$, then all arising constructions are permutations of the van der Corput sequence (counting in binary and reflected about the comma): \textit{greedy energy minimization recovers the way we count in binary.} This gives a new construction of the classical van der Corput sequence. The special case $f(x)=1-\log (2\sin (\pi x))$ answers a question of Steinerberger. Interestingly, the point sets we derive are also known in a different context as Leja sequences on the unit disk. Moreover, we give a general bound on the discrepancy of any sequence constructed in this way for functions $f$ satisfying an additional assumption.

18 pages, 7 figures, discrepancy bound added