  In this note we study the control problem for the heat equation on
$\mathbb{R}^d$, $d\geq 1$, with control set $\omega\subset\mathbb{R}^d$. We
provide a necessary and sufficient condition (called $(\gamma,
a)$-\emph{thickness}) on $\omega$ such that the heat equation is
null-controllable in any positive time. We give an estimate of the control cost
with explicit dependency on the characteristic geometric parameters of the
control set. Finally, we derive a control cost estimate for the heat equation
on cubes with periodic, Dirichlet, or Neumann boundary conditions, where the
control sets are again assumed to be thick. We show that the control cost
estimate is consistent with the $\mathbb{R}^d$ case.
