measure).

Let $S$ be an open subset of $\mathbb{R}^3$, $\delta \in (0,1)$, $\pmb{x}_0 \in \mathbb{R}^3$, and $r \in (0, \infty)$. $S$ is [*3D $\delta$-sparse around $\pmb{x}_0$ at scale $r$*]{} if $$\frac{m^3\bigl(S\cap B(\pmb{x}_0,r)\bigr)}{m^3\bigl(B(\pmb{x}_0,r)\bigr)} \leq \delta.$$

The main idea presented in [@Gr13] was to utilize sparseness of the vorticity super-level sets via the *harmonic measure maximum principle* (for the essentials on the harmonic measure in the complex plane see, e.g., [@Ahl]); in short, the sparseness translates into smallness of the harmonic measure associated with the regions of the intense vorticity which–in turn–provides a bound on the $L^\infty$-norm [^4], controlling a possible finite time blow-up. A key PDE technique utilized here was deriving the lower bounds on the radius of spatial analyticity of the solution in the $L^\infty$-type spaces. The principal result obtained in [@Gr13] states that as long as the suitably defined vorticity super-level sets are sparse at the scale comparable to the radius of spatial analyticity measured in $L^\infty$, no finite time blow-up can occur. Since the local-in-time spatially analytic smoothing is simply a (strong) manifestation of the diffusion, this result is consistent with the physics of turbulent dissipation.

As noted in the Introduction, the Navier-Stokes regularity problem is *supercritical*, i.e., 