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

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] \rightarrow $\mathbb{R}$$be(i)symmetric$f(x) = f(1-x)$,(ii)twicedifferentiableon$(0,1)$,and(iii)suchthat$f’'(x)>0$forall$x \in (0,1)$. We study the greedy dynamical system, where, given an initial set ${x_0, \ldots, x_{N-1}} \subset [0,1)$,thepoint$x_N$isobtainedas$$ x_{N} = \arg\min_{x \in [0,1)} \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.