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-
points” sit inside zero sets of harmonic polynomials in
of degree
(for all
and
) 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-
points ()
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
. An
application is given to the two-phase free boundary regularity problem for
harmonic measure below the continuous threshold introduced by Kenig and
Toro.
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