Hausdorff dimension of measures with arithmetically restricted spectrum

Abstract

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 those bounds via adaptation of our previous results for martingales on \(q\)-regular trees to a specific backwards martingale. To show the sharpness of our method, we improve the best numerical lower bound known for the Hausdorff dimension of certain Riesz products.