This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets.
Let u be a real-valued harmonic function in \mathbb{R}^n with u(0)=0 and n\geq 3. We prove \mathcal{H}^{n-1}(\{u=0\} \cap B(0,2)) \gtrsim_{\varepsilon} N^{1-\varepsilon}, where the doubling index N is a notion of growth defined by \sup_{B(0, 1)}|u| = 2^N \sup_{B(0,\frac{1}{2})}|u|. This gives an almost sharp lower bound for the Hausdorff measure of the zero set of u, which is conjectured to be linear in N. The new ingredients of the article are the notion of stable growth, and a multi-scale induction technique for a lower bound for the distribution of the doubling index of harmonic functions. It gives a significant improvement over the previous best-known bound \mathcal{H}^{n-1}\left(\{u=0\} \cap 2B\right)\geq \exp (c \log N/\log\log N ),
which implied Nadirashvili's conjecture.
