The Nadirashvili’s conjecture on the volume of the zero sets of harmonic functions is true

You need to know: Eucledean space ${\mathbb R}^n$ , origin $0$ in ${\mathbb R}^n$ , norm $||x||=\sqrt{\sum_{i=1}^nx_i^2}$ of $x=(x_1,\dots,x_n)\in{\mathbb R}^n$ , unit ball $B_n(0,1) =\{ x \in {\mathbb R}^n\,:\,||x||<1\}$ , open subset V of ${\mathbb R}^n$ , twice continuously differentiable function f on V, notation $\frac{\partial^2 f}{\partial x_i^2}$ for second-order partial derivatives.

Background: Let V be an open subset of ${\mathbb R}^n$ . A function $f: V \to {\mathbb R}$ is called harmonic if $\frac{\partial^2 f}{\partial x_1^2} + \dots + \frac{\partial^2 f}{\partial x_n^2} = 0$ for every $x\in V$ . A diameter of set $U\subset {\mathbb R}^n$ is $\text{diam} U=\sup\limits_{x,y \in U}||x-y||$ . For $\delta>0$ , a $\delta$ -cover of set $S \subset {\mathbb R}^n$ is a sequence of sets $U_1, U_2, \dots, U_i, \dots$ of diameters $\text{diam}U_i\leq \delta$ for all i such that $S \subset \bigcup\limits_{i=1}^\infty U_i$ . For $d\geq 0$ , let $H_\delta^d(S) = \inf \sum\limits_{i=1}^{\infty}(\text{diam}U_i)^d$ , where the infimum is taken over all $\delta$ -covers of S. The number $H^d(S)=\lim\limits_{\delta\to 0} H_\delta^d(S)$ is called the d-dimensional Hausdorff measure of S.

The Theorem: On 9th May 2016, Alexander Logunov submitted to arxiv a paper in which he proved that for every dimension $n\geq 3$ , there exists a constant $c=c(n)>0$ , depending only on n, such that inequality $H^{n-1}(\{f = 0\} \cap B(0,1)) \geq c$ holds for every harmonic function $f: B(0,1)\to {\mathbb R}$ such that $f(0)=0$ .

Short context: Harmonic functions are studied in many areas of mathematics, such as mathematical physics and the theory of stochastic processes. The Hausdorff measure $H^{n-1}$ is (up to a constant factor) just the usual $n-1$ -dimensional volume, e.g. $H^2$ is the area. The Theorem confirms a conjecture of Nadirashvili made in 1997. In particular, it implies that the zero set of any non-constant harmonic function $h:{\mathbb R}^3 \to {\mathbb R}$ has an infinite area Also, the Theorem is an important step towards proving a more general conjecture of Yau, which predicts a similar result on n-dimensional curved spaces (called “ $C^\infty$ -smooth Riemannian manifolds”) in place of ${\mathbb R}^n$ .