A note on weighted simultaneous Diophantine approximation on manifolds
Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 775-794

In this note we present an improvement to a recent result due to Beresnevich, Levesley, and Ward (2021) pertaining to weighted simultaneous Diophantine approximation on manifolds.

Dans cette note nous présentons une amélioration d’un résultat récent dû à Beresnevich, Levesley, et Ward (2021) sur l’approximation diophantienne simultanée pondérée sur les variétés.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1263
Classification : 11J13, 28A78, 11J83
Keywords: Weighted Diophantine approximation, Manifolds, Hausdorff dimension, Mass Transference Principle.

Allen, Demi 1 ; Wang, Baowei 2

1 Department of Mathematics and Statistics University of Exeter Harrison Building, North Park Road Exeter, EX4 4QF, United Kingdom
2 School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan, 430074, P. R. China
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Allen, Demi; Wang, Baowei. A note on weighted simultaneous Diophantine approximation on manifolds. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 775-794. doi: 10.5802/jtnb.1263

[1] Allen, Demi; Baker, Simon A general mass transference principle, Sel. Math., New Ser., Volume 25 (2019) no. 3, 39, 38 pages | Zbl | MR

[2] Allen, Demi; Beresnevich, Victor A mass transference principle for systems of linear forms and its applications, Compos. Math., Volume 154 (2018) no. 5, pp. 1014-1047 | DOI | MR | Zbl

[3] Beresnevich, Victor Rational points near manifolds and metric Diophantine approximation, Ann. Math., Volume 175 (2012) no. 1, pp. 187-235 | DOI | MR | Zbl

[4] Beresnevich, Victor; Dickinson, Detta; Velani, Sanju Diophantine approximation on planar curves and the distribution of rational points, Ann. Math., Volume 166 (2007) no. 2, pp. 367-426 | MR | Zbl | DOI

[5] Beresnevich, Victor; Lee, Lawrence; Vaughan, Robert C.; Velani, Sanju Diophantine approximation on manifolds and lower bounds for Hausdorff dimension, Mathematika, Volume 63 (2017) no. 3, pp. 762-779 | Zbl | DOI | MR

[6] Beresnevich, Victor; Levesley, Jason; Ward, Ben A lower bound for the Hausdorff dimension of the set of weighted simultaneously approximable points over manifolds, Int. J. Number Theory, Volume 17 (2021) no. 8, pp. 1795-1814 | Zbl | DOI | MR

[7] Beresnevich, Victor; Vaughan, Robert C.; Velani, Sanju; Zorin, Evgeniy Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory, Int. Math. Res. Not., Volume 2017 (2017) no. 10, pp. 2885-2908 | MR | Zbl

[8] Beresnevich, Victor; Velani, Sanju A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures, Ann. Math., Volume 164 (2006) no. 3, pp. 971-992 | DOI | MR | Zbl

[9] Beresnevich, Victor; Velani, Sanju Schmidt’s theorem, Hausdorff measures, and slicing, Int. Math. Res. Not., Volume 2006 (2006) no. 19, 48794, 24 pages | MR | Zbl

[10] Beresnevich, Victor; Velani, Sanju A note on simultaneous Diophantine approximation on planar curves, Math. Ann., Volume 337 (2007) no. 4, pp. 769-796 | DOI | MR | Zbl

[11] Beresnevich, Victor; Zorin, Evgeniy Explicit bounds for rational points near planar curves and metric Diophantine approximation, Adv. Math., Volume 225 (2010) no. 6, pp. 3064-3087 | DOI | MR | Zbl

[12] Duffin, Richard J.; Schaeffer, Albert C. Khintchine’s problem in metric Diophantine approximation, Duke Math. J., Volume 8 (1941), pp. 243-255 | MR | Zbl

[13] Falconer, Kenneth Fractal geometry: Mathematical foundations and applications, John Wiley & Sons, 2003, xxviii+337 pages | DOI

[14] Jarník, Vojtěch Über die simultanen diophantischen Approximationen, Math. Z., Volume 33 (1931) no. 1, pp. 505-543 | DOI | Zbl

[15] Khintchine, Aleksandr Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann., Volume 92 (1924), pp. 115-125 | DOI | Zbl

[16] Khintchine, Aleksandr Über die angenäherte Auflösung linearer Gleichungen in ganzen Zahlen, Rec. Math. Moscou, Volume 32 (1925), pp. 203-218

[17] Khintchine, Aleksandr Zur metrischen Theorie der diophantischen Approximationen, Math. Z., Volume 24 (1926) no. 1, pp. 706-714 | MR | Zbl | DOI

[18] Koivusalo, Henna; Rams, Michał Mass transference principle: from balls to arbitrary shapes, Int. Math. Res. Not., Volume 2021 (2021) no. 8, pp. 6315-6330 | DOI | MR | Zbl

[19] Koukoulopoulos, Dimitris; Maynard, James On the Duffin-Schaeffer conjecture, Ann. Math., Volume 192 (2020) no. 1, pp. 251-307 | Zbl | MR

[20] Pollington, Andrew D.; Vaughan, Robert C. The k-dimensional Duffin and Schaeffer conjecture, Mathematika, Volume 37 (1990) no. 2, pp. 190-200 | DOI | MR | Zbl

[21] Rudin, Walter Principles of mathematical analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, 1976, x+342 pages

[22] Rynne, Bryan P. Hausdorff dimension and generalized simultaneous Diophantine approximation, Bull. Lond. Math. Soc., Volume 30 (1998) no. 4, pp. 365-376 | DOI | MR | Zbl

[23] Schmidt, Wolfgang M. Diophantine approximation, Lecture Notes in Mathematics, 785, Springer, 1980, x+299 pages

[24] Simmons, David Some manifolds of Khinchin type for convergence, J. Théor. Nombres Bordeaux, Volume 30 (2018) no. 1, pp. 175-193 | Zbl | DOI | MR | Numdam

[25] Sprindžuk, Vladimir Metric theory of Diophantine approximations, Scripta Series in Mathematics, John Wiley & Sons, 1979, xiii+156 pages | MR

[26] Vaughan, Robert C.; Velani, Sanju Diophantine approximation on planar curves: the convergence theory, Invent. Math., Volume 166 (2006) no. 1, pp. 103-124 | DOI | MR | Zbl

[27] Wang, Baowei; Wu, Jun Mass transference principle from rectangles to rectangles in Diophantine approximation, Math. Ann., Volume 381 (2021) no. 1-2, pp. 243-317 | DOI | MR | Zbl

[28] Wang, Baowei; Wu, Jun; Xu, Jian Mass transference principle for limsup sets generated by rectangles, Math. Proc. Camb. Philos. Soc., Volume 158 (2015) no. 3, pp. 419-437 | DOI | MR | Zbl

[29] Zhong, Wenmin Mass transference principle: from balls to arbitrary shapes: measure theory, J. Math. Anal. Appl., Volume 495 (2021) no. 1, 124691, 23 pages | MR | Zbl

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