{'fraction_non_alphanumeric': 0.06608478802992519, 'fraction_numerical': 0.012468827930174564, 'mean_word_length': 4.425675675675675, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Continuity of the Hausdorff Measure of Continued Fractions and Countable Alphabet Iterated Function Systems

We prove that if by Jn(G) we denote the set of all numbers in [0, 1] whose infinite continued fraction expansions have all entries in the finite set {1, 2, . . . , n}, then limn→∞ Hhn(Jn(G))= 1 = H1(J(G)), where hn is the Hausdorff dimension of Jn(G), Hhn is the corresponding Hausdorff measure, and J(G) denotes the set of all irrational numbers in [0, 1], i .e. those whose continued fraction expansion is infinite. We also show that this property is not too common by constructing a class of infinite iterated function systems S on [0, 1], consisting of similarities, for which limF →E HhF (JF ) < HhS (JS); the lower limit is taken over finite subsets of the countable infinite alphabet E.