Abstract: A real number $x$ is essentially non-normal in base $b$ if each of of its digits $0,1,\ldots,b-1$ does not have a frequency. It was shown by Barreira, Saussol, and Schmeling in 2002 by using the thermodynamic formalism and in 2005 by Albeverio, Pratsiovytyi, Mykola, and Torbin using different methods that the set of essentially non-normal numbers in base $b$ has full Hausdorff dimension. Additionally, this set is comeagre.
We wish to provide the following type of strengthening of this result:
\item We will require the all blocks of digits and not just individual digits do not have a frequency.
\item We will additionally ask that blocks of digits do not have a frequency when sampled along arithmetic progressions and possibly some other polynomial sequences.
\item We will calculate the Hausdorff dimension of such sets in a large class of Cantor series expansions that include $b$-ary expansions as a special case.
In particular, for basic sequences $Q$ that are generic points of weakly mixing dynamical systems, we are able to show that the set of essentially non-normal numbers in the sense of the first two conditions is of full Hausdorff dimension and is comeagre, greatly generalizing previous results. Additionally, we show some weaker versions of these conditions for $Q$ that are translated Thue-Morse sequences which hold also along other polynomially indexed sequences. Our theorem will also apply to Sturmian sequences.
In addition to computing the Hausdorff dimension of sets of essentially non-normal numbers, we are able to provide computable examples of such numbers and effective presentations (in the sense that they lie in the lightface Borel hierarchy) of fractals of arbitrary large Hausdorff dimension consisting of numbers with these properties. Additionally, for some $Q$ we may fully effectively exhibit examples of fractals of arbitrarily large packing dimension of numbers which are all transcendental and satisfy many previously mentioned essential non-normality properties.