Fractional-matrix invariance of Diophantine systems of linear forms

V. G. Zhuravlev

Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 4, Tome 502 (2021), pp. 5-31

Résumé

It is known that under linear fractional unimodular transformations $\alpha \mapsto \alpha'= \frac{a \alpha + b} {c \alpha + d}$ the real numbers $\alpha $ and $\alpha'$ keep their expansions in the usual continued fractions up to a finite number of initial incomplete quotients. For this reason, these numbers have the same approximation speeds by their convergent fractions. This result is generalized to $(l \times k)$-matrices $ \alpha $. It is proved, if $ \alpha \mapsto \alpha'= (A \alpha + B)\cdot(C \alpha + D)^{- 1}$ for some fractional matrix unimodular transformation, then matrices $ \alpha $ and $ \alpha'$ have the same approximation speeds too. To prove this result we used the $\mathcal{L}$-algorithm based on the method of localizing units in algebraic number fields.