Improved bounds for the variant of lazy cops and robbers on generalized hypercubes

In the speed-$d$ variant of Lazy Cops and Robbers, the cops and the robber alternate turns. On a cop turn, either all cops remain stationary or one cop traverses a path of length at most $d$. On a robber turn, the robber either remains stationary or moves to an adjacent vertex. Let $c_{\mathrm L}^{(d)}(G)$ denote the minimum number of cops that can force a cop to occupy the robber's vertex after finitely many turns. We study this variant on the generalized hypercube $Q(n,m)$, whose vertex set is ${\{0,1,\ldots,m\}}^n$. For fixed integers $m\geq2$ and $d\geq1$, we prove that, as $n\to\infty$, \[ c_{\mathrm L}^{(d)}(Q(n,m)) =O\!\left(\frac{{(m+1)}^n}{n^{d+1/2}}\right). \] When $d=1$, our result improves the upper bound of Sim, Tan, and Wong for the ordinary lazy cop number by a factor of $\log n$.