  We provide a variety of lower bounds for the well-known shortcut set problem:
how much can one decrease the diameter of a directed graph on $n$ vertices and
$m$ edges by adding $O(n)$ or $O(m)$ of shortcuts from the transitive closure
of the graph. Our results are based on a vast simplification of the recent
construction of Bodwin and Hoppenworth [FOCS 2023] which was used to show an
$\widetilde{\Omega}(n^{1/4})$ lower bound for the $O(n)$-sized shortcut set
problem. We highlight that our simplification completely removes the use of the
convex sets by B\'ar\'any and Larman [Math. Ann. 1998] used in all previous
lower bound constructions. Our simplification also removes the need for
randomness and further removes some log factors. This allows us to generalize
the construction to higher dimensions, which in turn can be used to show the
following results. For $O(m)$-sized shortcut sets, we show an $\Omega(n^{1/5})$
lower bound, improving on the previous best $\Omega(n^{1/8})$ lower bound. For
all $\varepsilon > 0$, we show that there exists a $\delta > 0$ such that there
are $n$-vertex $O(n)$-edge graphs $G$ where adding any shortcut set of size
$O(n^{2-\varepsilon})$ keeps the diameter of $G$ at $\Omega(n^\delta)$. This
improves the sparsity of the constructed graph compared to a known similar
result by Hesse [SODA 2003].
  We also consider the sourcewise setting for shortcut sets: given a graph
$G=(V,E)$, a set $S\subseteq V$, how much can we decrease the sourcewise
diameter of $G$, $\max_{(s, v) \in S \times V, \text{dist}(s, v) < \infty}
\text{dist}(s,v)$ by adding a set of edges $H$ from the transitive closure of
$G$? We show that for any integer $d \ge 2$, there exists a graph $G=(V, E)$ on
$n$ vertices and $S \subseteq V$ with $|S| = \widetilde{\Theta}(n^{3/(d+3)})$,
such that when adding $O(n)$ or $O(m)$ shortcuts, the sourcewise diameter is
$\widetilde{\Omega}(|S|^{1/3})$.
