Geodesic
Fractional-matrix invariance of Diophantine systems of linear forms
Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 4, Tome 502 (2021), pp. 5-31 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {Fractional-matrix invariance of {Diophantine} systems of linear forms},
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V. G. Zhuravlev. Fractional-matrix invariance of Diophantine systems of linear forms. Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 4, Tome 502 (2021), pp. 5-31. http://geodesic.mathdoc.fr/item/ZNSL_2021_502_a0/

[1] V. G. Zhuravlev, “$\mathcal{L}$-algoritm approksimatsii diofantovykh sistem lineinykh form”, Algebra i analiz, 490 (2020), 25–48

[2] A. Ya. Khinchin, Tsepnye drobi, Chetvertoe izd., Nauka, M., 1978, 112 pp.

[3] V. Shmidt, Diofantovy priblizheniya, Mir, M., 1983, 228 pp.

[4] V. G. Zhuravlev, “Lokalizovannye matritsy Pizo i sovmestnye priblizheniya algebraicheskikh chisel”, Zap. nauch. semin. POMI, 458, 2017, 104–134

[5] V. G. Zhuravlev, “Diofantovy priblizheniya lineinykh form”, Algebra i analiz, 490 (2020), 5–24

[6] T. W. Cusick, “Diophantine Approximation of Ternary Linear Forms”, Mathematics of computation, 25:113 (1971), 163–180 | DOI | MR | Zbl

[7] T. W. Cusick, “Diophantine Approximation of Ternary Linear Forms. II”, Mathematics of computation, 26:120 (1972), 977–993 | DOI | MR | Zbl

[8] Z. I. Borevich, I. R. Shafarevich, Teoriya chisel, Trete izd., Nauka, M., 1985

[9] V. G. Zhuravlev, “Simpleks-modulnyi algoritm razlozheniya algebraicheskikh chisel v mnogomernye tsepnye drobi”, Zap. nauch. semin. POMI, 449, 2016, 168–195

[10] V. G. Zhuravlev, “Simpleks-yadernyi algoritm razlozheniya v mnogomernye tsepnye drobi”, Sovremennye problemy matematiki, 299, MIAN, 2017, 1–20

[11] V. G. Zhuravlev, “Drobno-lineinaya invariantnost mnogomernykh tsepnykh drobei”, Zap. nauch. semin. POMI, 458, 2017, 42–76

[12] V. G. Zhuravlev, “Drobno-lineinaya invariantnost simpleks-modulnogo algoritma razlozheniya algebraicheskikh chisel v mnogomernye tsepnye drobi”, Zap. nauch. semin. POMI, 458, 2017, 77–103