Krasikov lagarias 2003 bounds difference inequalities

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theorem_2_2: A feasible solution of the linear program L_k^NT(lambda), 1 <= lambda <= 2, attached to Krasikov's difference inequalities mod 3^k bounds every function phi_k^m(y) below by c_k^m lambda^y over four times the largest principal variable, although the inequalities contain advanced variables.

theorem_6_1: For each positive a not divisible by 3, at least x^0.84 of the integers n <= x have a in their 3x+1 orbit once x >= x_0(a); the proof is computer-aided, a feasible solution of the linear program for k = 11.

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Ilia Krasikov and Jeffrey C. Lagarias, Bounds for the 3x+1 Problem using Difference Inequalities, arXiv:math/0205002v1 (30 April 2002; 21 pp.); published Acta Arith. 109 (2003), no. 3, 237--258, DOI 10.4064/aa109-3-4.

Studies the systems of difference inequalities that Krasikov introduced in 1989 (the paper's [4]) for the occupancy of congruence classes mod $3^k$ under backward iteration, and shows (Theorem 2.2, p. 5) that the linear programs $L_k^{NT}(\lambda)$ attached to the original systems give valid lower bounds although those systems contain advanced variables. Theorem 6.1 (p. 16): for each positive $a\not\equiv0\pmod3$, the number of $n\le x$ whose forward orbit contains $a$ is at least $x^{0.84}$ for all $x\ge x_0(a)$; the proof is computer-aided, a feasible solution of $L_{11}^{NT}(\lambda)$ with $\lambda=1.7922310$ computed by D. Applegate. With $a=1$ this is the lower bound $\pi_1(x)\ge x^{0.84}$ for the count of integers below x whose orbit reaches 1, improving the bound $x^{0.81}$ of Applegate and Lagarias that p. 2 calls the best previous one. Relevance: proves the lower bound x^0.84 for the number of integers below x whose 3x+1 orbit reaches 1 (problem 1135).

The copy read for this card is arXiv v1. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0205002), every other right reserved.

Read status. Claims checked: Theorems 2.2 and 6.1 and the definitions they use (pp. 1--5, 15--16) were read on the print; the proofs of Theorems 3.1, 3.2, 4.1 and 5.1 and the computed feasible solution behind Table 2 were not checked.

Bears on. #1135: the map $T$ of the paper is the problem's $f$, and Theorem 6.1 with $a=1$ gives $\pi_1(x)\ge x^{0.84}$ for all large $x$, a lower bound on how many starting values up to $x$ reach $1$; it does not settle the problem.

Results. Theorem 2.2 (p. 5), the lower bound from feasible solutions of $L_k^{NT}(\lambda)$; Theorem 6.1 (p. 16), the bound $\pi_a(x)\ge x^{0.84}$.

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