2019/05/23 by Pausinger, Florian
#11B83 #11K31 #31C15 #49S05 #52C25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1905.09641
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 (i) symmetric f(x) = f(1-x), (ii) twice differentiable on (0,1), and (iii) 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(2 sin(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. Moreover, we give a general bound on the discrepancy of any sequence constructed in this way for functions f satisfying an additional assumption.