It is known that under linear-fractional unimodular transformations \( \upalpha \mapsto {\upalpha}^{\prime }=\frac{a\upalpha +b}{c\upalpha +d} \), the continuedfraction expansions of the real numbers α and α′ coincide up to a finite number of initial incomplete quotients. For this reason, the rates of approximation of these numbers by their convergents of continued fractions are the same. This result is generalized to l × k matrices. It is proved that if α ↦ α′ = (Aα + B) ・ (Cα + D)−1 is a unimodular matrix-fractional transformation, then the approximation rates for the matrices α and α′ are the same. The proof of this result uses the L - algorithm, which is based on a method for localizing units of algebraic number fields.
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References
V. G. Zhuravlev, “L-algorithm for approximating Diophantine systems of linear forms,” Algebra Anal., 490, 25–48 (2020).
A. Ya. Khinchin, Continued Fractions [in Russian], Nauka, Moscow (1978).
W. M. Schmidt, Diophantine Approximation [Russian translation], Mir, Moscow (1983).
V. G. Zhuravlev, “Localized Pizot matrices and joint approximations of algebraic numbers,” Zap. Nauchn. Semin POMI, 458, 104–134 (2017); English transl., J. Math. Sci., 234, 659–679 (2018).
V. G. Zhuravlev, “Diophantine approximations of linear forms,” Algebra Anal., 490, 5–24 (2020).
T. W. Cusick, “Diophantine approximation of ternary linear forms,” Math. Comput., 25, No. 113, 163–180 (1971).
T. W. Cusick, “Diophantine approximation of ternary linear forms. II,” Math. Comput., 26, No. 120, 977–993 (1972).
Z. I. Borevich and I. R. Shafarevich, Number Theory, 3rd Ed. [in Russian], Nauka, Moscow (1985).
V. G. Zhuravlev, “Simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions,” Zap. Nauchn. Semin. POMI, 449, 130–167 (2016); English transl., J. Math. Sci., 225, 924–949 (2017).
V. G. Zhuravlev, “Simplex-karyon algorithm for expansion in multidemensional continued fractions expansion,” Trudy MIAN, 299, 1–20 (2017).
V. G. Zhuravlev, “Linear-fractional invariance of multidimensional continued fractions,” Zap. Nauchn. Semin. POMI, 458, 42–76 (2017); English transl., J. Math. Sci., 234, 616–639 (2018).
V. G. Zhuravlev, “Linear-fractional invariance of the simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions,” Zap. Nauchn. Semin. POMI, 458, 77–103 (2017); English transl., J. Math. Sci., 234, 640–658 (2018).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 502, 2021, pp. 5–31.
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Zhuravlev, V.G. Matrix-Fractional Invariance of Diophantine Systems of Linear Forms. J Math Sci 264, 103–121 (2022). https://doi.org/10.1007/s10958-022-05983-w
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DOI: https://doi.org/10.1007/s10958-022-05983-w