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Let consider a subset of set of non-normal numbers for which a frequency does not exist for any individual digit. Such numbers called essentially non-normal numbers in base $b$. It was proven by Albeverio, Pratsiovytyi and Torbin in 2005 that this set has full Hausdorff dimension and is of second Baire category.  We extend and generalize this result for large class of Cantor series expansion considering numbers for which frequency does not exist for any block of digits for any length. Furthermore the result still holds for the set of essentially non-normal numbers whose Cantor series digits are sampled along all arithmetic progressions.
Talk based on a joint work with Dylan Ayrey, William Mance and Clemens Müllner.