Height gap conjectures, D-finiteness, and a weak dynamical Mordell–Lang conjecture

• Published in: Mathematische annalen Vol. 378; no. 3-4; pp. 971 - 992
• Main Authors: Bell, Jason P., Hu, Fei, Satriano, Matthew
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2020; Springer Nature B.V
• Subjects: Combinatorial analysis; Density; Mathematics; Mathematics and Statistics; Number theory; Power series; Sequences; 37P55; 14E05; 11G50
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-020-02062-w

In previous work, the first author, Ghioca, and the third author introduced a broad dynamical framework giving rise to many classical sequences from number theory and algebraic combinatorics. Specifically, these are sequences of the form f ( Φ n ( x ) ) , where Φ : X ⤏ X and f : X ⤏ P 1 are rational maps defined over Q ¯ and x ∈ X ( Q ¯ ) is a point whose forward orbit avoids the indeterminacy loci of Φ and f . They conjectured that if the sequence is infinite, then lim sup h ( f ( Φ n ( x ) ) ) log n > 0 . They also made a corresponding conjecture for lim inf and showed that it implies the Dynamical Mordell–Lang Conjecture. In this paper, we prove the lim sup conjecture as well as the lim inf conjecture away from a set of density 0. As applications, we prove results concerning the height growth rate of coefficients of D -finite power series as well as the Dynamical Mordell–Lang Conjecture up to a set of density 0.