Disjoint superheavy subsets and fragmentation norms

We present a lower bound for a fragmentation norm and construct a bi-Lipschitz embedding $I\colon \mathbb{R}^n\to\mathrm{Ham}(M)$ with respect to the fragmentation norm on the group $\mathrm{Ham}(M)$ of Hamiltonian diffeomorphisms of a symplectic manifold $(M,\omega)$. As an application, we provide an answer to Brandenbursky's question on fragmentation norms on $\mathrm{Ham}(\Sigma_g)$, where $\Sigma_g$ is a closed Riemannian surface of genus $g\geq 2$