A lower bound for periods of matrices

Pietro Corvaja, Zéev Rudnick, Umberto Zannier

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)535-541
Number of pages7
JournalCommunications in Mathematical Physics
Volume252
Issue number1-3
DOIs
StatePublished - Dec 2004

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