Structure of sets which are well approximated by zero sets of harmonic polynomials

TL;DR

This paper investigates the structure and dimensional properties of zero sets of harmonic polynomials, providing a comprehensive framework for understanding their singular points and approximations, with implications for free boundary problems.

Contribution

It offers a general structure theorem for sets approximated by harmonic polynomial zero sets, including sharp dimension estimates without PDE techniques.

Findings

Dimension estimates for singular sets depend on the degree and topology.

Sets with good local approximation have a well-understood structure.

Results apply to free boundary regularity in harmonic measure problems.

Abstract

The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries a detailed study of the singular points of these zero sets is required. In this paper we study how "degree $k$ points" sit inside zero sets of harmonic polynomials in $\mathbb R^n$ of degree $d$ (for all $n\geq 2$ and $1\leq k\leq d$) and inside sets that admit arbitrarily good local approximations by zero sets of harmonic polynomials. We obtain a general structure theorem for the latter type of sets, including sharp Hausdorff and Minkowski dimension estimates on the singular set of "degree $k$ points" ($k\geq 2$) without proving uniqueness of blowups or aid of PDE methods such as monotonicity formulas. In addition, we show that in the presence of a certain topological separation condition, the sharp dimension estimates improve and depend on the parity of $k$. An application is given to the two-phase free boundary regularity problem for harmonic measure below the continuous threshold introduced by Kenig and Toro.