  An oblivious subspace embedding (OSE), characterized by parameters
$m,n,d,\epsilon,\delta$, is a random matrix $\Pi\in \mathbb{R}^{m\times n}$
such that for any $d$-dimensional subspace $T\subseteq \mathbb{R}^n$,
$\Pr_\Pi[\forall x\in T, (1-\epsilon)\|x\|_2 \leq \|\Pi x\|_2\leq
(1+\epsilon)\|x\|_2] \geq 1-\delta$. When an OSE has $s\le 1/2.001\epsilon$
nonzero entries in each column, we show it must hold that $m =
\Omega\left(d^2/( \epsilon^2s^{1+O(\delta)})\right)$, which is the first lower
bound with multiplicative factors of $d^2$ and $1/\epsilon$, improving on the
previous $\Omega\left(d^2/s^{O(\delta)}\right)$ lower bound due to Li and Liu
(PODS 2022). When an OSE has $s=\Omega(\log(1/\epsilon)/\epsilon)$ nonzero
entries in each column, we show it must hold that $m =
\Omega\left((d/\epsilon)^{1+1/4.001\epsilon s}/s^{O(\delta)}\right)$, which is
the first lower bound with multiplicative factors of $d$ and $1/\epsilon$,
improving on the previous $\Omega\left(d^{1+1/(16\epsilon s+4)}\right)$ lower
bound due to Nelson and Nguyen (ICALP 2014). This second result is a special
case of a more general trade-off among $d,\epsilon,s,\delta$ and $m$.
