  The geometry of solutions to the higher dimensional Einstein vacuum equations
presents aspects that are absent in four dimensions, one of the most remarkable
being the existence of stably trapped null geodesics in the exterior of
asymptotically flat black holes. This paper investigates the stable trapping
phenomenon for two families of higher dimensional black holes, namely black
strings and black rings, and how this trapping structure is responsible for the
slow decay of linear waves on their exterior. More precisely, we study decay
properties for the energy of solutions to the scalar, linear wave equation
$\Box_{g_{\textup{ring}}} \Psi=0$, where $g_{\textup{ring}}$ is the metric of a
fixed black ring solution to the five-dimensional Einstein vacuum equations.
For a class $\mathfrak{g}$ of black ring metrics, we prove a logarithmic lower
bound for the uniform energy decay rate on the black ring exterior
$(\mathcal{D},g_{\textup{ring}})$, with $g_{\textup{ring}}\in\mathfrak{g}$. The
proof generalizes the perturbation argument and quasimode construction of
Holzegel--Smulevici \cite{SharpLogHolz} to the case of a non-separable wave
equation and crucially relies on the presence of stably trapped null geodesics
on $\mathcal{D}$. As a by-product, the same logarithmic lower bound can be
established for any five-dimensional black string.
  Our result is the first mathematically rigorous statement supporting the
expectation that black rings are dynamically unstable to generic perturbations.
In particular, we conjecture a new \textit{nonlinear} instability for
five-dimensional black strings and thin black rings which is already present at
the level of scalar perturbations and clearly differs from the mechanism driven
by the well-known Gregory--Laflamme instability.
