Garcia tal 1999 official

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M. V. P. Garcia and F. A. Tal, A note on the generalized 3n+1 problem, Acta Arith. 90 (1999) 245-250, DOI 10.4064/aa-90-3-245-250. Six pages, A4, 211,308 bytes, PDF 1.2 (the journal digitization, dvipdfm production).

Relevance: for the Hasse map $H$ of (1) (p. 245), with $m>d\ge2$, $\gcd(d,m)=1$, $m<d^{d/(d-1)}$ and $R_d$ the system of residues modulo $d$ fixed there, Theorem 1 (p. 249) shows that a complete set of representatives of the relation on the positive integers given by $a\sim b$ iff $H^k(a)=H^k(b)$ for some $k\in\mathbb{N}$ has Banach density zero, and Corollary 1 (p. 249) that every orbit $\mathcal{O}(n)$ under $H$ has Banach density zero. For $d=2$ the hypotheses force $m=3$, and with $R_2=\{0,-1\}$ the map $H$ is the Collatz shortcut map of problem 1135 (p. 245). Statements read on the page images of pp. 245 and 249; proofs not checked.

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