Skip to main content
We gratefully acknowledge support from the Simons Foundation, member institutions, and all contributors. Donate
arXiv logo
Cornell University Logo

Mathematics > Number Theory

arXiv:math/0309215 (math)
[Submitted on 12 Sep 2003 (v1), last revised 9 Feb 2004 (this version, v3)]

Title:A lower bound for periods of matrices

Authors:Pietro Corvaja, Zeev Rudnick, Umberto Zannier
View PDF
Abstract: For a nonsingular integer matrix A, we study the growth of the order of A modulo N. We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field.
For exceptional matrices, it is easily seen that there are arbitrarily large values of N for which the order of A modulo N is logarithmically small. In contrast, we show that if the matrix is not exceptional, then the order of A modulo N goes to infinity faster than any constant multiple of log N.
Comments: Added references and corrected a few misprints. Added condition that A be ergodic for a remark in the introduction
Subjects: Number Theory (math.NT); Mathematical Physics (math-ph); Chaotic Dynamics (nlin.CD)
MSC classes: 11A07; 11Jxx
Cite as: arXiv:math/0309215 [math.NT]
  (or arXiv:math/0309215v3 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.math/0309215
Related DOI: https://doi.org/10.1007/s00220-004-1184-6

Submission history

From: Zeev Rudnick [view email]
[v1] Fri, 12 Sep 2003 15:04:15 UTC (8 KB)
[v2] Fri, 12 Dec 2003 14:38:03 UTC (9 KB)
[v3] Mon, 9 Feb 2004 16:56:14 UTC (9 KB)
Full-text links:

Access Paper:

view license
Current browse context:
math.NT
< prev   |   next >
new | recent | 2003-09

References & Citations

export BibTeX citation

Bookmark

BibSonomy logo Reddit logo

Bibliographic and Citation Tools

Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)

Code, Data and Media Associated with this Article

alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)

Demos

Replicate (What is Replicate?)
Hugging Face Spaces (What is Spaces?)
TXYZ.AI (What is TXYZ.AI?)

Recommenders and Search Tools

Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)