  This paper is concerned with quantitative estimates for the Navier-Stokes
equations.
  First we investigate the relation of quantitative bounds to the behaviour of
critical norms near a potential singularity with Type I bound
$\|u\|_{L^{\infty}_{t}L^{3,\infty}_{x}}\leq M$. Namely, we show that if $T^*$
is a first blow-up time and $(0,T^*)$ is a singular point then
$$\|u(\cdot,t)\|_{L^{3}(B_{0}(R))}\geq
C(M)\log\Big(\frac{1}{T^*-t}\Big),\,\,\,\,\,\,R=O((T^*-t)^{\frac{1}{2}-}).$$ We
demonstrate that this potential blow-up rate is optimal for a certain class of
potential non-zero backward discretely self-similar solutions.
  Second, we quantify the result of Seregin (2012), which says that if $u$ is a
smooth finite-energy solution to the Navier-Stokes equations on
$\mathbb{R}^3\times (0,1)$ with
$$\sup_{n}\|u(\cdot,t_{(n)})\|_{L^{3}(\mathbb{R}^3)}<\infty\,\,\,\textrm{and}\,\,\,t_{(n)}\uparrow
1,$$ then $u$ does not blow-up at $t=1$.
  To prove our results we develop a new strategy for proving quantitative
bounds for the Navier-Stokes equations. This hinges on local-in-space smoothing
results (near the initial time) established by Jia and \v{S}ver\'{a}k (2014),
together with quantitative arguments using Carleman inequalities given by Tao
(2019).
  Moreover, the technology developed here enables us in particular to give a
quantitative bound for the number of singular points in a Type I blow-up
scenario.
