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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
DOI: https://doi.org/10.1090/S0002-9939-2013-11740-5
Published electronically: September 26, 2013
MathSciNet review: 3119183
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Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [B] Robert L. Benedetto, Preperiodic points of polynomials over global fields, J. Reine Angew. Math. 608 (2007), 123-153. MR 2339471 (2008j:11071), https://doi.org/10.1515/CRELLE.2007.055
  • [CCGHSZ] M.-C. Chang, J. Cilleruelo, M. Garaev, J. Hernandez, I. Shparlinski, A. Zumalacarregui, Points on curves in small boxes and applications (preprint).
  • [C] M. Chang, Factorization in generalized arithmetic progressions and applications to the Erdős-Szemerédi sum-product problems, Geom. Funct. Anal. 13 (2003), no. 4, 720-736. MR 2006555 (2004g:11007), https://doi.org/10.1007/s00039-003-0428-5
  • [CGOS] Javier Cilleruelo, Moubariz Z. Garaev, Alina Ostafe, and Igor E. Shparlinski, On the concentration of points of polynomial maps and applications, Math. Z. 272 (2012), no. 3-4, 825-837. MR 2995141, https://doi.org/10.1007/s00209-011-0959-7
  • [GS] Jaime Gutierrez and Igor E. Shparlinski, Expansion of orbits of some dynamical systems over finite fields, Bull. Aust. Math. Soc. 82 (2010), no. 2, 232-239. MR 2685147 (2011j:11153), https://doi.org/10.1017/S0004972709001270

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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: https://doi.org/10.1090/S0002-9939-2013-11740-5
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

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