Abstract
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 \(f(\Phi ^n(x))\), where \(\Phi :X\dasharrow X\) and \(f:X\dasharrow {\mathbb {P}}^1\) are rational maps defined over \(\overline{{\mathbb {Q}}}\) and \(x\in X(\overline{{\mathbb {Q}}})\) is a point whose forward orbit avoids the indeterminacy loci of \(\Phi \) and f. They conjectured that if the sequence is infinite, then \(\limsup \frac{h(f(\Phi ^n(x)))}{\log n} > 0\). They also made a corresponding conjecture for \(\liminf \) and showed that it implies the Dynamical Mordell–Lang Conjecture. In this paper, we prove the \(\limsup \) conjecture as well as the \(\liminf \) 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.
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Notes
Generalizing Definition 3.1, the upper asymptotic (or natural) density of \({\mathbf {T}}\subseteq {\mathbb {N}}^m\) is defined by \(\overline{d}({\mathbf {T}}) :=\limsup _{n\rightarrow \infty } |{\mathbf {T}}\cap [0, n]^m|/(n+1)^m\).
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Acknowledgements
We would like to thank Ken Davidson, Vesselin Dimitrov, Vern Paulsen, and Joe Silverman for helpful conversations. We are also grateful to the referee for fruitful suggestions and comments.
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Communicated by Kannan Soundararajan.
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The authors were partially supported by Discovery Grants from the National Science and Engineering Research Council of Canada.
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Bell, J.P., Hu, F. & Satriano, M. Height gap conjectures, D-finiteness, and a weak dynamical Mordell–Lang conjecture. Math. Ann. 378, 971–992 (2020). https://doi.org/10.1007/s00208-020-02062-w
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DOI: https://doi.org/10.1007/s00208-020-02062-w