A celebrated theorem of Lind states that a positive real number is equal to the entropy of a shift of finite type, if and only if, it is equal to the logarithm of a Perron algebraic integer divided by some natural number n. Given a Perron number p and a natural number n, we prove that there is a non-negative integral irreducible matrix with spectral radius equal to the nth root of p, and with dimension bounded above in terms of n, the algebraic degree, the spectral ratio, and certain arithmetic information about the ring of integers of its number field. Consequently, there is an irreducible shift of finite type with entropy equal to the logarithm of p divided by n, and with `size' bounded above in terms of the aforementioned data.

Reference:
- M. Yazdi, Non-negative integral matrices with given spectral radius and controlled dimension, arXiv:2101.09268 (2021).

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