Abstract
The paper is devoted to the study of fractal properties of subsets of the set of non-normal numbers with respect to Rényi f-expansions generated by continuous increasing piecewise linear functions defined on [0,+∞). All such expansions are expansions for real numbers generated by infinite linear IFS f={f0,f1,…,fn,…} with the following list of ratios Q∞=(q0,q1,…,qn,…).We prove the superfractality of the set of Q∞-essentially non-normal numbers, i.e. real numbers having no asymptotic frequencies of any digits from the alphabet A={0,1,…,n,…}, for any infinite stochastic vector Q∞, independently of the finiteness resp. infiniteness of its entropy and independently of the faithfulness resp. non-faithfulness of the family of cylinders generated by these expansions.
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