Bell lagarias 2014 genfun natural boundaries

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theorem_1_1: For the 3x+k map with k congruent to 1 or -1 mod 6, the generating function over the positive integers of a finite union of backward orbits is rational exactly when, beyond some point, the union consists of the integers in a set of residue classes mod |k|, and that set is then closed under r to 2r and r to 3r.

theorem_1_2: For the 3x+1 map, the backward-orbit generating function of every starting value m >= 1 other than 1, 2, 4 and 8 has the unit circle as natural boundary, and for those four values it is rational if the 3x+1 conjecture is true and has the unit circle as natural boundary if it is false.

theorem_1_3: For the 3x-1 map on the positive integers, the backward-orbit generating function of every starting value m >= 1 has the unit circle as natural boundary, which proves a conjecture of Berg and Opfer.

theorem_1_4: For the 3x+k map with k congruent to 1 or -1 mod 6, the backward-orbit generating function over the positive integers has the unit circle as natural boundary for all but finitely many starting values m >= 1.

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Jason P. Bell and Jeffrey C. Lagarias, 3x+1 Inverse Orbit Generating Functions Almost Always Have Natural Boundaries, arXiv:1408.6884v1 (2014; published Acta Arith. 170 (2015) 101-120; 15 pp.).

For the 3x+k map with k = +-1 mod 6, Theorem 1.1 (p. 3) characterizes when the generating function of a finite union of backward orbits, restricted to the positive integers, is rational: exactly when that union, from some point on, consists of the integers in a set of residue classes mod |k|, a set then closed under r -> 2r and r -> 3r. Its proof uses the Skolem-Mahler-Lech theorem; Theorems 1.2 to 1.4 combine it with the Polya-Carlson dichotomy (a power series with integer coefficients and radius of convergence 1 is rational or has the unit circle as natural boundary). Theorem 1.2 (p. 4): for k = 1 the function f_{1,m} has the unit circle as natural boundary for every m >= 1 except possibly m = 1, 2, 4, 8, unconditionally; for those four it is rational if the 3x+1 conjecture holds and has the natural boundary if it fails. Theorem 1.3 (p. 4) gives the natural boundary for every m >= 1 for the 3x-1 map, proving a conjecture of Berg and Opfer, and Theorem 1.4 (p. 4) for all but finitely many m >= 1 for general k. Read status: claims checked; the four theorems were read clause by clause on the PDF page images and their proofs for structure only.

Source: PDF. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1408.6884), every other right reserved.

Bears on.

• #1135: the problem's map is the paper's T_1 and its question is the paper's 3x+1 Conjecture. Theorem 1.2 (2) restates the question as whether f_{1,1}(z) (equally f_{1,2}, f_{1,4} or f_{1,8}) is a rational function; part (1) holds unconditionally. The paper does not decide the problem.

Results.

• Theorem 1.1 (p. 3): rationality of the generating function of a finite union of 3x+k backward orbits, characterized by residue classes mod |k|.
• Theorem 1.2 (p. 4): natural boundary for f_{1,m}, m >= 1, except possibly m = 1, 2, 4, 8, whose rationality follows from the 3x+1 conjecture and whose natural boundary follows from its failure.
• Theorem 1.3 (p. 4): natural boundary for f_{-1,m} for every m >= 1.
• Theorem 1.4 (p. 4): natural boundary for f_{k,m} for all but finitely many m >= 1.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.