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A lower bound for periods of matrices. (English) Zbl 1124.11306

Summary: 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 \(\log N\).

MSC:

11C20 Matrices, determinants in number theory
11J13 Simultaneous homogeneous approximation, linear forms
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