Equidistribution rate for Fekete points on some real manifolds

TL;DR

This paper establishes the rate of convergence for Fekete points to the equilibrium measure on certain real manifolds, extending previous results to more general real submanifolds using Cauchy-Riemann geometry techniques.

Contribution

It extends the known convergence rate results for Fekete points to real submanifolds with specific geometric conditions, using new geometric methods.

Findings

Convergence rate estimate holds for Fekete points on certain real manifolds.

Results apply to compact sets in R^n and the unit sphere in R^{n+1}.

Techniques involve Cauchy-Riemann geometry.

Abstract

Let L be a positive line bundle over a compact complex projective manifold X and K be a compact subset of X which is regular in a sense of pluripotential theory. A Fekete configuration of order k is a finite subset of K maximizing a Vandermonde type determinant associated with the power L^k of L. Berman, Boucksom and Witt Nystr\"om proved that the empirical measure associated with a Fekete configuration converges to the equilibrium measure of K as k tends to infinity. Dinh, Ma and Nguyen obtained an estimate for the rate of convergence. Using techniques from Cauchy-Riemann geometry, we show that the last result holds when K is a real nondegenerate C^5-piecewise submanifold of X such that its tangent space at any regular point is not contained in a complex hyperplane of the tangent space of X at that point. In particular, the estimate holds for Fekete points on some compact sets in R^n or the unit sphere in R^{n+1}.