The set of non-normal numbers for a multiplicative semigroup

We investigate the Hausdorff and packing dimensions of exceptional sets of numbers that are not normal with respect to a given multiplicative semigroup (in the sense of equidistribution). Our main results are twofold. First, we show that if the semigroup has only finitely many prime factors, then the corresponding exceptional set has full Hausdorff dimension. Second, we demonstrate that even with infinitely many prime factors, the exceptional set can still attain full packing dimension, provided the prime factors increase sufficiently rapidly.