A note on discreteness of $F$-jumping numbers

Suppose that $R$ is a ring essentially of finite type over a perfect field of characteristic $p>0$ and that $a\subseteq R$ is an ideal. We prove that the set of $F$-jumping numbers of ${\tau }_b(R;a^t)$ has no limit points under the assumption that $R$ is normal and $Q$-Gorenstein – we do \emph{not} assume that the $Q$-Gorenstein index is not divisible by $p$. Furthermore, we also show that the $F$-jumping numbers of ${\tau }_b(R;\mathrm{\Delta },a^t)$ are discrete under the more general assumption that $K_R+\mathrm{\Delta }$ is $\text{bR}$-Cartier.