Hausdorff dimension for level sets of upper and lower limits of generalized averages of binary digits

The problem of averaging of binary digits of numbers in [0, 1] is considered. A class (mu)of Toeplitz matrices regular with respect to usual (Cesaro) averages is characterized. The Hausdorff dimension of the level sets of the upper and lower limits of some generalized averages is explicitly computed and it is proved to be equal for every T in (mu). A description of sets on which finite measures on [0, 1] are concentrated is given using Toeplitz matrices in (mu).