Korec 1994 density estimate

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theorem_1: For every real c greater than log_4 3, about 0.7925, the set of nonnegative integers y with T^n(y) < y^c for some n, where T is the shortcut 3x+1 map, has asymptotic density 1.

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Ivan Korec, A density estimate for the 3x+1 problem, Math. Slovaca 44 (1994) 85-89 (DML-CZ digitized copy, dml.cz/dmlcz/133225; 5 pp. plus repository cover).

Read status: claims checked for the setting and Theorem 1 (p. 85), the remarks on p. 86, the notation and Lemmas 1--2 (p. 86), read clause by clause on the page images; the proof (pp. 87--88) and Examples 1--2 (pp. 88--89) were read for structure only and not checked.

Works with the shortcut map T on the nonnegative integers. For every real c > log_4 3 (= 0.79248125...), the set M_c of initial values y with T^n(y) < y^c for some n has asymptotic density 1 (Theorem 1, p. 85). The proof combines Terras's periodicity theorem (the parity vectors of the first m steps of x and y agree exactly when x and y are congruent mod 2^m; Lemma 1, p. 86) with a central-limit count: for every d > 1/2, the share of 0 <= y < 2^m with at most md odd steps among the first m tends to 1 (Lemma 2, p. 86). Examples 1--2 (pp. 88--89) show that the theorem does not follow at once from Terras's bounded-time density result and that lowering the bound on c may be nontrivial. Relevance: proves that for every c > log_4 3 the set of y whose 3x+1 trajectory drops below y^c has density 1; it does not decide problem 1135.

Source: PDF. The file prints "© Mathematical Institute of the Slovak Academy of Sciences, 1994" on its DML-CZ cover page (PDF p. 1), with terms of use that grant access to the digitized copy "strictly for personal use", and "©1994 Mathematical Institute Slovak Academy of Sciences" on PDF p. 2, every other right reserved.

Bears on. #1135: the paper's T, defined on the nonnegative integers, is given by the same formula as the problem's f; Theorem 1 shows that for each c > log_4 3 the starting values whose orbit falls below y^c have asymptotic density 1, and says nothing about whether every orbit reaches 1.

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