Lattice points to the sphere: towards discrepancy estimates

Description

Coauthor(s): Josef Dick, Yuan Xu
Special session: Periodic Point Configurations and Lattice Point Interactions p.64
It is a well-known fact that an N -point configuration on the unit sphere in R3 that maximizes the sum of all mutual Euclidean distances has minimal spherical cap L2-discrepancy (Stolarsky’s Invariance Principle).
Bounds for the maximal sum of distances show that this discrepancy tends to 0 as N → ∞ with convergence rate N^(-3/4) . The precise asymptotic behavior is closely related to unresolved questions about the asymptotic expansion of optimal Riesz s-energy and, in turn, universal optimality of planar configuration in the context of best-packing, renormalized energy, and the optimality of the hexagonal lattice.
For constructible point sets on the sphere less is known.
Using the area-preserving Lambert cylindrical equal-area projection, a planar configuration like a (rational) lattice can be mapped to the sphere.
The so far best possible provable bound for the L2 -discrepancy is the same as for the expected value for i.i.d. random points from 2012 for which the rate of convergence is N^(-1/2) .
In this talk, we present recent (partial) results for the Fibonacci lattice mapped to the sphere and comment on hyperuniformity of such configurations.

Period	18 Jul 2022
Event title	15th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing: MCQMC 2022
Event type	Conference
Location	Linz, AustriaShow on map
Degree of Recognition	International