  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.
