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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Expansions of quadratic maps in prime fields


Author: Mei-Chu Chang
Journal: Proc. Amer. Math. Soc. 142 (2014), 85-92
MSC (2010): Primary 11B50, 37A45, 11B75; Secondary 11T23, 37F10, 11G99
Published electronically: September 26, 2013
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Abstract: Let $ f(x)=ax^2+bx+c\in \mathbb{Z}[x]$ be a quadratic polynomial with $ a\not \equiv 0$ mod $ p$. Take $ z\in \mathbb{F}_p$ and let $ \mathcal O_z=\{f_i(z)\}_{i\in \mathbb{Z}^+}$ be the orbit of $ z$ under $ f$, where $ f_i(z)=f(f_{i-1}(z))$ and $ f_0(z)=z$. For $ M< \vert\mathcal O_z\vert$, we study the diameter of the partial orbit $ \mathcal O_M=\{z, f(z), f_2(z),\dots , f_{M-1}(z)\}$ and prove that there exists $ c_1>0$ such that

$\displaystyle \text {\rm diam } \mathcal O_M \gtrsim \min \bigg \{ Mp^{\;{c_1}}... ...rac 1{\log p} M^{\frac 45} p^{\frac 15}, M^{\;\frac 1{13}\log \log M}\bigg \}. $

For a complete orbit $ \mathcal C$, we prove that

$\displaystyle \text {\rm diam } \mathcal C \gtrsim \min \{p^{\;5c_1}, e^{ \;T/4}\;\},$

where $ T$ is the period of the orbit.

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Additional Information

Mei-Chu Chang
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: mcc@math.ucr.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11740-5
PII: S 0002-9939(2013)11740-5
Keywords: Dynamical system, orbits, additive combinatorics, exponential sums
Received by editor(s): October 10, 2011
Received by editor(s) in revised form: March 2, 2012
Published electronically: September 26, 2013
Additional Notes: The author’s research was partially financed by the National Science Foundation.
Communicated by: Bryna Kra
Article copyright: © Copyright 2013 American Mathematical Society