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 ’welldistributed’ 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.
Original language  English 

Pages (fromto)  165 
Journal  Annali di Matematica Pura ed Applicata 
Volume  200 
Early online date  18 May 2020 
DOIs  
Publication status  Early online date  18 May 2020 
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Florian Pausinger
Person: Academic