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

Abstract

AbstractThis 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 $$f:[0,1] \rightarrow {\mathbb {R}}$$ f : [ 0 , 1 ] → R be (1) symmetric $$f(x) = f(1-x)$$ f ( x ) = f ( 1 - x ) , (2) twice differentiable on (0, 1), and (3) such that $$f''(x)>0$$ f ′ ′ ( x ) > 0 for all $$x \in (0,1)$$ x ∈ ( 0 , 1 ) . We study the greedy dynamical system, where, given an initial set $$\{x_0, \ldots , x_{N-1}\} \subset [0,1)$$ { x 0 , … , x N - 1 } ⊂ [ 0 , 1 ) , the point $$x_N$$ x N is obtained as $$\begin{aligned} x_{N} = \arg \min _{x \in [0,1)} \sum _{k=0}^{N-1}{f(|x-x_k|)}. \end{aligned}$$ x N = arg min x ∈ [ 0 , 1 ) ∑ k = 0 N - 1 f ( | x - x k | ) . We prove that if we start this construction with the single element $$x_0=0$$ x 0 = 0 , then all arising constructions are permutations of the van der Corput sequence (counting in binary and reflected about the comma): 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 $$f(x) = 1-\log (2 \sin (\pi x))$$ f ( x ) = 1 - log ( 2 sin ( π 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.