  We consider the growth of the norms of transfer matrices of ergodic discrete
Schr\"odinger operators in one dimension. It is known that the set of energies
at which the rate of exponential growth is slower than prescribed by the
Lyapunov exponent is residual in the part of the spectrum at which the Lyapunov
exponent is positive. On the other hand, this exceptional set is of vanishing
Hausdorff measure with respect to any gauge function $\rho(t)$ such that
$\rho(t)/t$ is integrable at zero. Here we show that this condition on
$\rho(t)$ can not in general be improved: for operators with independent,
identically distributed potentials of sufficiently regular distribution, the
set of energies at which the rate of exponential growth is arbitrarily slow has
infinite Hausdorff measure with respect to any gauge function $\rho(t)$ such
that $\rho(t)/t$ is non-increasing and not integrable at zero. The main
technical ingredient, possibly of independent interest, is a Jarn\'ik-type
theorem describing the Hausdorff measure of the set of real numbers well
approximated by the eigenvalues of the Schr\"odinger operator. The proof of
this result relies on the theory of Anderson localisation and on the mass
transference principle of Beresnevich--Velani.
