A remark on the set of exactly approximable vectors in the simultaneous case
HTML articles powered by AMS MathViewer
- by Reynold Fregoli;
- Proc. Amer. Math. Soc. 152 (2024), 3177-3182
- DOI: https://doi.org/10.1090/proc/16790
- Published electronically: June 14, 2024
- HTML | PDF | Request permission
Abstract:
We compute the Hausdorff dimension of the set of $\psi$-exactly approximable vectors, in the simultaneous case, in dimension strictly larger than $2$ and for approximating functions $\psi$ with order at infinity less than or equal to $-2$. Our method relies on the analogous result in dimension $1$, proved by Yann Bugeaud and Carlos Moreira, and a version of Jarník’s theorem on fibres.References
- J. D. Bovey and M. M. Dodson, The Hausdorff dimension of systems of linear forms, Acta Arith. 45 (1986), no. 4, 337–358. MR 847294, DOI 10.4064/aa-45-4-337-358
- Victor Beresnevich, Detta Dickinson, and Sanju Velani, Sets of exact ‘logarithmic’ order in the theory of Diophantine approximation, Math. Ann. 321 (2001), no. 2, 253–273. MR 1866488, DOI 10.1007/s002080100225
- Victor Beresnevich, Detta Dickinson, and Sanju Velani, Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc. 179 (2006), no. 846, x+91. MR 2184760, DOI 10.1090/memo/0846
- A. S. Besicovitch, Sets of Fractional Dimensions (IV): On Rational Approximation to Real Numbers, J. London Math. Soc. 9 (1934), no. 2, 126–131. MR 1574327, DOI 10.1112/jlms/s1-9.2.126
- Prasuna Bandi, Anish Ghosh, and Debanjan Nandi, Exact approximation order and well-distributed sets, Adv. Math. 414 (2023), Paper No. 108871, 19. MR 4540922, DOI 10.1016/j.aim.2023.108871
- Yann Bugeaud and Carlos Gustavo Moreira, Sets of exact approximation order by rational numbers III, Acta Arith. 146 (2011), no. 2, 177–193. MR 2747026, DOI 10.4064/aa146-2-5
- Victor Beresnevich, Felipe Ramírez, and Sanju Velani, Metric Diophantine approximation: aspects of recent work, Dynamics and analytic number theory, London Math. Soc. Lecture Note Ser., vol. 437, Cambridge Univ. Press, Cambridge, 2016, pp. 1–95. MR 3618787
- Yann Bugeaud, Sets of exact approximation order by rational numbers, Math. Ann. 327 (2003), no. 1, 171–190. MR 2006007, DOI 10.1007/s00208-003-0445-6
- Victor Beresnevich and Sanju Velani, Schmidt’s theorem, Hausdorff measures, and slicing, Int. Math. Res. Not. , posted on (2006), Art. ID 48794, 24. MR 2264714, DOI 10.1155/IMRN/2006/48794
- Victor Beresnevich and Sanju Velani, Classical metric Diophantine approximation revisited: the Khintchine-Groshev theorem, Int. Math. Res. Not. IMRN 1 (2010), 69–86. MR 2576284, DOI 10.1093/imrn/rnp119
- Kenneth Falconer, Fractal geometry, 3rd ed., John Wiley & Sons, Ltd., Chichester, 2014. Mathematical foundations and applications. MR 3236784
- Robert Fraser and Reuben Wheeler, Fourier dimension estimates for sets of exact approximation order: the well-approximable case, Int. Math. Res. Not. IMRN 24 (2023), 20943–20969. MR 4681277, DOI 10.1093/imrn/rnac256
- P. X. Gallagher, Metric simultaneous diophantine approximation. II, Mathematika 12 (1965), 123–127. MR 188154, DOI 10.1112/S0025579300005234
- V. Jarník, Diophantische Approximationen und Hausdorffsches Maß, Rec. Math. Moscou 36 (1929), 371–382.
- Vojtěch Jarník, Über die simultanen diophantischen Approximationen, Math. Z. 33 (1931), no. 1, 505–543 (German). MR 1545226, DOI 10.1007/BF01174368
- A. Khintchine, Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann. 92 (1924), no. 1-2, 115–125 (German). MR 1512207, DOI 10.1007/BF01448437
- Felipe A. Ramírez, David S. Simmons, and Fabian Süess, Rational approximation of affine coordinate subspaces of Euclidean space, Acta Arith. 177 (2017), no. 1, 91–100. MR 3589916, DOI 10.4064/aa8521-8-2016
Bibliographic Information
- Reynold Fregoli
- Affiliation: Department of Mathematics, University of Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
- Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- MR Author ID: 1318722
- Email: reynoldfregoli@gmail.com
- Received by editor(s): April 5, 2023
- Received by editor(s) in revised form: December 20, 2023, and January 6, 2024
- Published electronically: June 14, 2024
- Additional Notes: The author was supported by SNSF grant 200021–182089.
- Communicated by: Rachel Pries
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3177-3182
- MSC (2020): Primary 11J13, 11J83
- DOI: https://doi.org/10.1090/proc/16790
- MathSciNet review: 4767253