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We consider the growth of the norms of transfer matrices of ergodic discreteSchr\"odinger operators in one dimension. It is known that the set of energiesat which the rate of exponential growth is slower than prescribed by theLyapunov exponent is residual in the part of the spectrum at which the Lyapunovexponent is positive. On the other hand, this exceptional set is of vanishingHausdorff 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, theset of energies at which the rate of exponential growth is arbitrarily slow hasinfinite Hausdorff measure with respect to any gauge function $\rho(t)$ suchthat $\rho(t)/t$ is non-increasing and not integrable at zero. The maintechnical ingredient, possibly of independent interest, is a Jarn\'ik-typetheorem describing the Hausdorff measure of the set of real numbers wellapproximated by the eigenvalues of the Schr\"odinger operator. The proof ofthis result relies on the theory of Anderson localisation and on the masstransference principle of Beresnevich--Velani.

On the Hausdorff measure of sets of non-Lyapunov behaviour, and a Jarnik-type theorem for random Schroedinger operators