Arkiv för Matematik

  • Ark. Mat.
  • Volume 47, Number 2 (2009), 243-266.

Diophantine exponents for mildly restricted approximation

Yann Bugeaud and Simon Kristensen

Full-text: Open access

Abstract

We are studying the Diophantine exponent μn, l defined for integers 1≤l< n and a vector α∈ℝn by letting $\mu_{n,l}=\sup\{\mu\geq0: 0 < \Vert\underline{x}\cdot\alpha\Vert<H(\underline{x})^{-\mu}\ \text{for infinitely many}\ \underline{x}\in\mathcal{C}_{n,l}\cap\mathbb{Z}^n\},$ where $\cdot$ is the scalar product, $\|\cdot\|$ denotes the distance to the nearest integer and $\mathcal{C}_{n,l}$ is the generalised cone consisting of all vectors with the height attained among the first l coordinates. We show that the exponent takes all values in the interval [l+1,∞), with the value n attained for almost all α. We calculate the Hausdorff dimension of the set of vectors α with μn, l(α)=μ for μ≥n. Finally, letting wn denote the exponent obtained by removing the restrictions on $\underline{x}$, we show that there are vectors α for which the gaps in the increasing sequence μn,1(α)≤...≤μn, n-1(α)≤wn(α) can be chosen to be arbitrary.

Article information

Source
Ark. Mat., Volume 47, Number 2 (2009), 243-266.

Dates
Received: 10 September 2007
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907080

Digital Object Identifier
doi:10.1007/s11512-008-0074-0

Mathematical Reviews number (MathSciNet)
MR2529700

Zentralblatt MATH identifier
1304.11062

Rights
2008 © Institut Mittag-Leffler

Citation

Bugeaud, Yann; Kristensen, Simon. Diophantine exponents for mildly restricted approximation. Ark. Mat. 47 (2009), no. 2, 243--266. doi:10.1007/s11512-008-0074-0. https://projecteuclid.org/euclid.afm/1485907080


Export citation

References

  • Beresnevich, V., On approximation of real numbers by real algebraic numbers, Acta Arith. 90 (1999), 97–112.
  • Beresnevich, V., Bernik, V. I., Kleinbock, D. Y. and Margulis, G. A., Metric Diophantine approximation: the Khintchine–Groshev theorem for nondegenerate manifolds, Mosc. Math. J. 2 (2002), 203–225.
  • Beresnevich, V., Dickinson, D. and Velani, S., Sets of exact ‘logarithmic’ order in the theory of Diophantine approximation, Math. Ann. 321 (2001), 253–273.
  • Beresnevich, V. and Velani, S., A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures, Ann. of Math. 164 (2006), 971–992.
  • Beresnevich, V. and Velani, S., Schmidt’s theorem, Hausdorff measures, and slicing, Int. Math. Res. Not. (2006), Art. ID 48794.
  • Bernik, V. I. and Dodson, M. M., Metric Diophantine Approximation on Manifolds, Cambridge Tracts in Mathematics 137, Cambridge University Press, Cambridge, 1999.
  • Bugeaud, Y., Mahler’s classification of numbers compared with Koksma’s, Acta Arith. 110 (2003), 89–105.
  • Bugeaud, Y., Approximation by Algebraic Numbers, Cambridge Tracts in Mathematics 160, Cambridge University Press, Cambridge, 2004.
  • Bugeaud, Y. and Laurent, M., On exponents of homogeneous and inhomogeneous Diophantine approximation, Mosc. Math. J. 5 (2005), 747–766, 972.
  • Cassels, J. W. S., An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics 45, Cambridge University Press, New York, 1957.
  • Davenport, H. and Schmidt, W. M., A theorem on linear forms, Acta Arith. 14 (1967/1968), 209–223.
  • Davenport, H. and Schmidt, W. M., Approximation to real numbers by algebraic integers, Acta Arith. 15 (1968/1969), 393–416.
  • Dickinson, D. and Velani, S. L., Hausdorff measure and linear forms, J. Reine Angew. Math. 490 (1997), 1–36.
  • Dodson, M. M., Geometric and probabilistic ideas in the metric theory of Diophantine approximations, Uspekhi Mat. Nauk 48:5 (1993), 77–106 (Russian). English transl.: Russian Math Surveys 48:5 (1993), 73–102.
  • Groshev, A. V., A theorem on systems of linear forms, Dokl. Akad. Nauk SSSR 19 (1938), 151–152.
  • Jarník, V., Über die simultanen diophantischen Approximationen, Math. Z. 33 (1931), 505–543.
  • Jarník, V., Über die angenäherte Lösung der Gleichung x1θ1+...+xnθn+x0=0 in ganzen Zahlen, Časopis Pěst Mat. Fys. 66 (1937), 192–205.
  • Koksma, J. F., Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen, Monatsh. Math. Phys. 48 (1939), 176–189.
  • Mahler, K., Zur Approximation der Exponentialfunktionen und des Logarithmus. I, II, J. Reine. Angew. Math. 166 (1932), 118–136; 137–150.
  • Rynne, B. P., The Hausdorff dimension of certain sets arising from Diophantine approximation by restricted sequences of integer vectors, Acta Arith. 61 (1992), 69–81.
  • Schmidt, W. M., T-numbers do exist, in Symposia Mathematica, Vol. IV (INdAM, Rome, 1968/69), pp. 3–26, Academic Press, London, 1970.
  • Schmidt, W. M., Mahler’s T-numbers, in 1969 Number Theory Institute (State Univ. New York, Stony Brook, NY, 1969), Proc. Sympos. Pure Math., 20, pp. 275–286, Amer. Math. Soc., Providence, RI, 1971.
  • Schmidt, W. M., Two questions in Diophantine approximation, Monatsh. Math. 82 (1976), 237–245.
  • Schmidt, W. M., Open problems in Diophantine approximation, in Diophantine Approximations and Transcendental Numbers (Luminy, 1982), Progr. Math. 31, pp. 271–287, Birkhäuser, Boston, MA, 1983.
  • Sprindzhuk, V. G., Mahler’s Problem in Metric Number Theory, Izdat. Nauka i Tekhnika, Minsk, 1967 (Russian). English transl.: Translations of Mathematical Monographs 25, Amer. Math. Soc., Providence, RI, 1969.
  • Sprindzhuk, V. G., Metric Theory of Diophantine Approximations, Winston, Washington, DC, 1979.
  • Thurnheer, P., Un raffinement du théorème de Dirichlet sur l’approximation diophantienne, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), 623–624.
  • Thurnheer, P., Eine Verschärfung des Satzes von Dirichlet über diophantische Approximation, Comment. Math. Helv. 57 (1982), 60–78.
  • Thurnheer, P., Zur diophantischen Approximation von zwei reellen Zahlen, Acta Arith. 44 (1984), 201–206.
  • Thurnheer, P., Approximation diophantienne par certains couples d’entiers, C. R. Math. Rep. Acad. Sci. Canada 7 (1985), 51–53.
  • Thurnheer, P., On Dirichlet’s theorem concerning Diophantine approximation, Acta Arith. 54 (1990), 241–250.