On zero sets in the Dirichlet space

TL;DR

This paper investigates the properties of zero sets in the Dirichlet space, introducing new families of zero sets and establishing conditions under which certain sets can be accumulation points of zeros.

Contribution

It provides new families of zero sets in the Dirichlet space and characterizes accumulation points based on logarithmic capacity, using Carleson formula techniques.

Findings

New families of zero sets identified

Any closed set with zero logarithmic capacity can be accumulation points

Zeros satisfy a growth restriction depending on the set

Abstract

We study the zeros sets of functions in the Dirichlet space. Using Carleson formula for Dirichlet integral, we obtain some new families of zero sets. We also show that any closed subset of $E \subset \TT$ with logarithmic capacity zero is the accumulation points of the zeros of a function in the Dirichlet space. The zeros satisfy a growth restriction which depends on $E$.