TY - JOUR

T1 - Greedy energy minimization can count in binary: point charges and the van der Corput sequence

AU - Pausinger, Florian

PY - 2020/5/18

Y1 - 2020/5/18

N2 - 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]→R be (1) symmetric f(x)=f(1−x), (2) twice differentiable on (0, 1), and (3) such that f′′(x)>0 for all x∈(0,1). We study the greedy dynamical system, where, given an initial set {x0,…,xN−1}⊂[0,1), the point xN is obtained as
xN=argminx∈[0,1)∑k=0N−1f(|x−xk|).
We prove that if we start this construction with the single element x0=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(2sin(πx)) 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.

AB - 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]→R be (1) symmetric f(x)=f(1−x), (2) twice differentiable on (0, 1), and (3) such that f′′(x)>0 for all x∈(0,1). We study the greedy dynamical system, where, given an initial set {x0,…,xN−1}⊂[0,1), the point xN is obtained as
xN=argminx∈[0,1)∑k=0N−1f(|x−xk|).
We prove that if we start this construction with the single element x0=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(2sin(πx)) 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.

U2 - 10.1007/s10231-020-00990-7

DO - 10.1007/s10231-020-00990-7

M3 - Article

JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

SN - 0373-3114

ER -