  We prove a logarithmic improvement of the Caffarelli-Kohn-Nirenberg partial
regularity theorem for the Navier-Stokes equations. The key idea is to find a
quantitative counterpart for the absolute continuity of the dissipation energy
using the pigeonhole principle. Based on the same method, for any suitable weak
solution, we show the existence of intervals of regularity in one spatial
direction with length depending exponentially on the natural local energies of
the solution. Then, we give two applications of the latter result in the
axially symmetric case. The first one is a local quantitative regularity
criterion for suitable weak solutions with small swirl. The second one is a
slightly improved one-point CKN criterion which implies all known (slightly
supercritical) Type I regularity results in the literature.
