Some remarks about level sets of Cesaro averages of binary digits

The problem of averaging the binary digits of numbers in $left[ 0,1 ight] $ is considered. It is well known that Lebesgue a.e. in $left[ 0,1 ight] $ the usual Cesaro average is equal to $rac{1}{2}$ and that the Hausdorff dimension of the set where the Cesaro average is equal to $alpha $ is given by an entropy function $dleft( alpha ight) $. We prove that if $alpha eq rac{1}{2}$ then the Hausdorff measure $mathcal{H}^{dleft( alpha ight) }$ of such set is infinite. We moreover explicitly construct an infinite matrix $T$ (in a class $mathcal{M}$ of Toeplitz matrices regular with respect to Cesaro averages) such that the Hausdorff dimension of the set of the points not having Cesaro average and where the $T$-generalized average is $alpha $ is still given by $dleft( alpha ight) $.