Yazdi, Mehdi Non-negative integral matrices with given spectral radius and controlled dimension. (English) Zbl 1509.37010 Ergodic Theory Dyn. Syst. 42, No. 10, 3246-3269 (2022). Summary: A celebrated theorem of D. A. Lind [Ergodic Theory Dyn. Syst. 4, 283–300 (1984; Zbl 0546.58035)] states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number \(p\), we prove that there is an integral irreducible matrix with spectral radius \(p\), and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number \(p\), there is an irreducible shift of finite type with entropy \(\log(p)\) defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data. MSC: 37A44 Relations between ergodic theory and number theory 37B10 Symbolic dynamics 15B36 Matrices of integers 15A18 Eigenvalues, singular values, and eigenvectors 15B48 Positive matrices and their generalizations; cones of matrices 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure Keywords:Perron-Frobenius theory; Perron numbers; shifts of finite type; entropy Citations:Zbl 0546.58035 PDFBibTeX XMLCite \textit{M. Yazdi}, Ergodic Theory Dyn. Syst. 42, No. 10, 3246--3269 (2022; Zbl 1509.37010) Full Text: DOI arXiv References: [1] Banaszczyk, W.. New bounds in some transference theorems in the geometry of numbers. Math. Ann.296(1) (1993), 625-635. · Zbl 0786.11035 [2] Bayer Fluckiger, E.. 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