On zero sets in the Dirichlet space

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 $E\subset \text{TT}$ 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 $E$.