Quantitative Diophantine approximation with congruence conditions
Journal de Théorie des Nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 261-271.

In this short paper we prove a quantitative version of the Khintchine–Groshev Theorem with congruence conditions. Our argument relies on a classical argument of Schmidt on counting generic lattice points, which in turn relies on a certain variance bound on the space of lattices.

Dans ce court article, nous prouvons une version quantitative du théorème de Khintchine–Groshev avec des conditions de congruence. Notre argument repose sur un argument classique de Schmidt sur le comptage de points de réseau génériques, qui à son tour repose sur une certaine borne de variance sur l’espace des réseaux.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1161
Classification: 11N56,  14G42
Mahbub Alam 1; Anish Ghosh 1; Shucheng Yu 2

1 School of Mathematics Tata Institute of Fundamental Research Mumbai 400005, India
2 Department of Mathematics Uppsala University, Box 480 SE-75106, Uppsala, Sweden
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Mahbub Alam; Anish Ghosh; Shucheng Yu. Quantitative Diophantine approximation with congruence conditions. Journal de Théorie des Nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 261-271. doi : 10.5802/jtnb.1161. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1161/

[1] Mahbub Alam; Anish Ghosh Equidistribution on homogeneous spaces and the distribution of approximates in Diophantine approximation, Trans. Am. Math. Soc., Volume 373 (2020) no. 5, pp. 3357-3374 | Article | MR: 4082241 | Zbl: 1445.11066

[2] Mahbub Alam; Anish Ghosh Quantitative rational approximation on spheres (2020) (https://arxiv.org/abs/2003.02243)

[3] Jayadev Athreya; Andrew Parrish; Jimmy Tseng Ergodic theory and Diophantine approximation for translation surfaces and linear forms, Nonlinearity, Volume 29 (2016) no. 8, pp. 2173-2190 | Article | MR: 3538408 | Zbl: 1365.37036

[4] Anish Ghosh; Dubi Kelmer; Shucheng Yu Effective Density for Inhomogeneous Quadratic Forms I: Generic Forms and Fixed Shifts, Int. Math. Res. Not. (2020), rnaa206 | Article

[5] Jens Marklof; Andreas Strömbergsson The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems, Ann. Math., Volume 172 (2010) no. 3, pp. 1949-2033 | Article | MR: 2726104 | Zbl: 1211.82011

[6] Erez Nesharim; René Rühr; Ronggang Shi Metric Diophantine approximation with congruence conditions, Int. J. Number Theory, Volume 16 (2020) no. 9, pp. 1923-1933 | Article | MR: 4153361 | Zbl: 1455.11096

[7] Wolfgang M. Schmidt A metrical theorem in diophantine approximation, Can. J. Math., Volume 12 (1960), pp. 619-631 | Article | MR: 118711 | Zbl: 0097.26205

[8] Wolfgang M. Schmidt A metrical theorem in geometry of numbers, Trans. Am. Math. Soc., Volume 95 (1960), pp. 516-529 | Article | MR: 117222 | Zbl: 0101.27904

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