Convergence of the distance squared Gibbs measure on algebraic sets, targets, and fractals

Abstract

The distance squared Gibbs measure is a certain probability measure on Euclidean space studied in [S] and [PS]. In this paper, we document what is known about the convergence of a sequence of such measures over certain mathematical spaces, including algebraic sets, fractals, and "targets." Specifically, we find that the converging measure will distribute evenly over subanalytic sets and fractals, but may not distribute evenly over other types of spaces, such as "targets," that is, circles of decreasing radii where the total length is infinite.