  We investigate the dimension-parametric complexity of the reachability
problem in vector addition systems with states (VASS) and its extension with
pushdown stack (pushdown VASS). Up to now, the problem is known to be
$\mathcal{F}_k$-hard for VASS of dimension $3k+2$ (the complexity class
$\mathcal{F}_k$ corresponds to the $k$th level of the fast-growing hierarchy),
and no essentially better bound is known for pushdown VASS. We provide a new
construction that improves the lower bound for VASS: $\mathcal{F}_k$-hardness
in dimension $2k+3$. Furthermore, building on our new insights we show a new
lower bound for pushdown VASS: $\mathcal{F}_k$-hardness in dimension $\frac k 2
+ 4$. This dimension-parametric lower bound is strictly stronger than the upper
bound for VASS, which suggests that the (still unknown) complexity of the
reachability problem in pushdown VASS is higher than in plain VASS (where it is
Ackermann-complete).
