Geodesic
On the transference principle and Nesterenko's linear independence criterion
Izvestiya. Mathematics , Tome 87 (2023) no. 2, pp. 252-264.

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We consider the problem of simultaneous approximation of real numbers $\theta_1, \dots,\theta_n$ by rationals and the dual problem of approximating zero by the values of the linear form $x_0+\theta_1x_1+\dots+\theta_nx_n$ at integer points. In this setting we analyse two transference inequalities obtained by Schmidt and Summerer. We present a rather simple geometric observation which proves their result. We also derive several previously unknown corollaries. In particular, we show that, together with German's inequalities for uniform exponents, Schmidt and Summerer's inequalities imply the inequalities by Bugeaud and Laurent and “one half” of the inequalities by Marnat and Moshchevitin. Moreover, we show that our main construction provides a rather simple proof of Nesterenko's linear independence criterion.
Keywords: Diophantine exponents, transference inequalities, linear independence criterion.
Mots-clés : Diophantine approximation 
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O. N. German; N. G. Moshchevitin. On the transference principle and Nesterenko's linear independence criterion. Izvestiya. Mathematics , Tome 87 (2023) no. 2, pp. 252-264. http://geodesic.mathdoc.fr/item/IM2_2023_87_2_a1/

[1] Yu. V. Nesterenko, “On the linear independence of numbers”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1985, no. 1, 46–49 ; English transl. Moscow Univ. Math. Bull., 40:1 (1985), 69–74 | MR | Zbl

[2] A. Khintchine, “Über eine Klasse linearer Diophantischer Approximationen”, Rend. Circ. Mat. Palermo, 50 (1926), 170–195 | DOI | Zbl

[3] V. Jarník, “Zum Khintchineschen “Übertragungssatz””, Trav. Inst. Math. Tbilissi, 3 (1938), 193–216 | Zbl

[4] Y. Bugeaud and M. Laurent, “Exponents of Diophantine approximation”, Diophantine geometry, CRM Series, 4, Ed. Norm., Pisa, 2007, 101–121 | MR | Zbl

[5] Y. Bugeaud and M. Laurent, “On transfer inequalities in Diophantine approximation. II”, Math. Z., 265:2 (2010), 249–262 | DOI | MR | Zbl

[6] O. N. German, “Intermediate Diophantine exponents and parametric geometry of numbers”, Acta Arith., 154:1 (2012), 79–101 | DOI | MR | Zbl

[7] O. N. German, “On Diophantine exponents and Khintchine's transference principle”, Mosc. J. Comb. Number Theory, 2:2 (2012), 22–51 | MR | Zbl

[8] W. M. Schmidt and L. Summerer, “Diophantine approximation and parametric geometry of numbers”, Monatsh. Math., 169:1 (2013), 51–104 | DOI | MR | Zbl

[9] O. N. German and N. G. Moshchevitin, “A simple proof of Schmidt–Summerer's inequality”, Monatsh. Math., 170:3-4 (2013), 361–370 | DOI | MR | Zbl

[10] V. Jarník, “Une remarque sur les approximations diophantiennes linéaires”, Acta Sci. Math. (Szeged), 12 B (1950), 82–86 | MR | Zbl

[11] V. Jarník, “Contribution à la théorie des approximations diophantiennes linéaires et homogènes”, Czechoslovak Math. J., 4:79 (1954), 330–353 | MR | Zbl

[12] A. Marnat and N. G. Moshchevitin, “An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation”, Mathematika, 66:3 (2020), 818–854 | DOI | MR

[13] Ngoc Ai Van Nguyen, A. Poëls, and D. Roy, “A transference principle for simultaneous rational approximation”, J. Théor. Nombres Bordeaux, 32:2 (2020), 387–402 | DOI | MR | Zbl

[14] J. Schleischitz, “On geometry of numbers and uniform rational approximation to the Veronese curve”, Period. Math. Hung., 83:2 (2021), 233–249 ; arXiv: 1911.03215 | DOI | MR | Zbl

[15] J. Schleischitz, “Optimality of two inequalities for exponents of Diophantine approximation”, J. Number Theory, 244 (2023), 169–203 ; arXiv: 2102.03154 | DOI | MR

[16] D. Kleinbock, N. Moshchevitin, and B. Weiss, “Singular vectors on manifolds and fractals”, Israel J. Math., 245:2 (2021), 589–613 ; arXiv: 1912.13070 | DOI | MR | Zbl

[17] S. Fischler and W. Zudilin, “A refinement of Nesterenko's linear independence criterion with applications to zeta values”, Math. Ann., 347:4 (2010), 739–763 | DOI | MR | Zbl

[18] A. Chantanasiri, “On the criteria for linear independence of Nesterenko, Fischler and Zudilin”, Chamchuri J. Math., 2:1 (2010), 31–46 | MR | Zbl

[19] S. Fischler and T. Rivoal, “Irrationality exponent and rational approximations with prescribed growth”, Proc. Amer. Math. Soc., 138:3 (2010), 799–808 | DOI | MR | Zbl