  We provide an estimate from below for the lower Hausdorff dimension of
measures on the unit circle based on the arithmetic properties of their
spectra. We obtain our bounds via application of a general result for abstract
$q$-regular martingales to the Gundy--Varopoulos backwards martingale. To show
the sharpness of our method, we improve the best known numerical lower bound
for the Hausdorff dimension of certain Riesz products.
