  We study the relation between propagation of smallness in the plane and
control for heat equations. The former has been proved by Zhu who showed how
the value of solutions in some small set propagates to a larger domain. By
reviewing his proof, we establish a quantitative version with the explicit
dependence of parameters. Using this explicit version, we establish new exact
null-controllability results of 1D heat equations with any nonnegative power
growth potentials $V\in\mathcal{C}(\mathbb{R})$. As a key ingredient, new
spectral inequalities are established. The control set $\Omega$ that we
consider satisfy \begin{equation*}
  \left|\Omega\cap [x-L\langle x\rangle ^{-s},x+L\langle x\rangle
^{-s}]\right|\ge \gamma^{\langle x\rangle^\tau}2L\langle x\rangle^{-s}
\end{equation*} for some $\gamma\in(0,1)$, $L>0$, $\tau,s\ge 0$, and $\langle
x\rangle:=(1+|x|^2)^{1 /2} $. In particular, the null-controllability result
for the case of thick sets that allow the decay of the density (\textit{i.e.},
$s=0$ and $\tau\ge 0$) is included. These extend the Zhu-Zhuge's results from
$\Omega$ being the union of equidistributive open sets to thick sets in the
1-dimensional case, and Su-Sun-Yuan's result from bounded potentials to certain
unbounded ones.
