{'fraction_non_alphanumeric': 0.012594458438287154, 'fraction_numerical': 0.0025188916876574307, 'mean_word_length': 4.527777777777778, 'pattern_counts': {'":': 0, '<': 0, '<?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}}
Universally meager sets

We study category counterparts of the notion of a universal measure zero set of reals. We say that a set A C R1 is universally meager if every Borel isomorphic image of A is meager in IR. We give various equivalent definitions emphasizing analogies with the universally null sets of reals. In particular, two problems emerging from an earlier work of Grzegorek are solved.