  In $d$ dimensions, accurately approximating an arbitrary function oscillating
with frequency $\lesssim k$ requires $\sim k^d$ degrees of freedom. A numerical
method for solving the Helmholtz equation (with wavenumber $k$ and in $d$
dimensions) suffers from the pollution effect if, as $k\to\infty$, the total
number of degrees of freedom needed to maintain accuracy grows faster than this
natural threshold (i.e., faster than $k^d$ for domain-based formulations, such
as finite element methods, and $k^{d-1}$ for boundary-based formulations, such
as boundary element methods).
  It is well known that the $h$-version of the finite element method (FEM)
(where accuracy is increased by decreasing the meshwidth $h$ and keeping the
polynomial degree $p$ fixed) suffers from the pollution effect, and research
over the last $\sim$ 30 years has resulted in a near-complete rigorous
understanding of how quickly the number of degrees of freedom must grow with
$k$ to maintain accuracy.
  In contrast to the $h$-FEM, at least empirically, the $h$-version of the
boundary element method (BEM) does $\textit{not}$ suffer from the pollution
effect (recall that in the boundary element method the scattering problem is
reformulated as an integral equation on the boundary of the scatterer, with
this integral equation then solved numerically using a finite-element-type
approximation space). However, the current best results in the literature on
how quickly the number of degrees of freedom for the $h$-BEM must grow with $k$
to maintain accuracy fall short of proving this.
  In this paper, we prove that the $h$-version of the Galerkin method applied
to the standard second-kind boundary integral equations for solving the
Helmholtz exterior Dirichlet problem does not suffer from the pollution effect
when the obstacle is nontrapping (i.e., does not trap geometric-optic rays).
