Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators


Author: J. C. Lagarias
Journal: Trans. Amer. Math. Soc. 272 (1982), 545-554
MSC: Primary 10F10; Secondary 10F20
MathSciNet review: 662052
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper defines the notion of a best simultaneous Diophantine approximation to a vector $ \alpha$ in $ R^n$ with respect to a norm $ \left\Vert \,\cdot\, \right\Vert $ on $ R^n$. Suppose $ \alpha$ is not rational and order the best approximations to $ \alpha$ with respect to $ \left\Vert\, \cdot\, \right\Vert $ by increasing denominators $ 1=q_1 < q_2 < \cdots$. It is shown that these denominators grow at least at the rate of a geometric series, in the sense that

$\displaystyle g\left( {\alpha ,\,\left\Vert {\,\cdot\,} \right\Vert} \right) = ... ...liminf\limits_{k \to \infty }} {({q_k})^{1/k}} \geq 1 + \frac{1}{{{2^{n + 1}}}}$

. Let $ g\left( {\left\Vert\, \cdot\, \right\Vert} \right)$ denote the infimum of $ g\left( {\alpha ,\,\left\Vert {\,\cdot\,} \right\Vert} \right)$ over all $ \alpha$ in $ R^n$ with an irrational coordinate. For the sup norm $ \left\Vert\, \cdot \,\right\Vert _s$ on $ R^2$ it is shown that $ g\left( {\left\Vert \, \cdot \, \right\Vert}_s \right)\ge\theta=1.270^{+}$ where $ \theta^4=\theta^{2}+1$.

References [Enhancements On Off] (What's this?)

  • [1] William W. Adams, Simultaneous diophantine approximations and cubic irrationals, Pacific J. Math. 30 (1969), 1–14. MR 0245522
  • [2] Arne J. Brentjes, A two-dimensional continued fraction algorithm for best approximations with an application in cubic number fields, J. Reine Angew. Math. 326 (1981), 18–44. MR 622343, 10.1515/crll.1981.326.18
  • [3] J. W. S. Cassels, Simultaneous Diophantine approximation, J. London Math. Soc. 30 (1955), 119–121. MR 0066432
  • [4] T. W. Cusick, Formulas for some Diophantine approximation constants. II, Acta Arith. 26 (1974/75), 117–128. MR 0354563
  • [5] T. W. Cusick, Best Diophantine approximation for ternary linear forms, J. Reine Angew. Math. 315 (1980), 40–52. MR 564522, 10.1515/crll.1980.315.40
  • [6] H. Davenport, Simultaneous Diophantine approximation, Proc. London Math. Soc. (3) 2 (1952), 406–416. MR 0054657
  • [7] H. Davenport, Simultaneous Diophantine approximation, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, Erven P. Noordhoff N.V., Groningen; North-Holland Publishing Co., Amsterdam, 1956, pp. 9–12. MR 0085302
  • [8] H. Davenport and K. Mahler, Simultaneous Diophantine approximation, Duke Math. J. 13 (1946), 105–111. MR 0016068
  • [9] H. Davenport and Wolfgang M. Schmidt, Approximation to real numbers by quadratic irrationals, Acta Arith. 13 (1967/1968), 169–176. MR 0219476
  • [10] H. Davenport and W. M. Schmidt, A theorem on linear forms, Acta Arith. 14 (1967/1968), 209–223. MR 0225728
  • [11] E. Dubois, $ F$-best approximation of zero by a cubic linear form: Calculation of the fundamental unit of a not totally real cubic field, Proc. Queen's Number Theory Conf., 1979 (P. Ribenboim, ed.), Queen's Papers in Pure and Appl. Math., no. 54, Queen's University, Kingston, 1980, pp. 205-222.
  • [12] Eugène Dubois and Georges Rhin, Approximations simultanées de deux nombres réels, Séminaire Delange-Pisot-Poitou, 20e année: 1978/1979. Théorie des nombres, Fasc. 1 (French), Secrétariat Math., Paris, 1980, pp. Exp. No. 9, 13 (French, with English summary). MR 582422
  • [13] W. Jurkat, W. Kratz, and A. Peyerimhoff, On best two-dimensional Dirichlet-approximations and their algorithmic calculation, Math. Ann. 244 (1979), no. 1, 1–32. MR 550059, 10.1007/BF01420334
  • [14] J. C. Lagarias, Some new results in simultaneous Diophantine approximation, Proc. Queen's Number Theory Conf., 1979 (P. Ribenboim, ed), Queen's Papers in Pure and Appl. Math., no. 54, Queens University, Kingston, 1980, pp. 453-474.
  • [15] J. C. Lagarias, Best simultaneous Diophantine approximations. II. Behavior of consecutive best approximations, Pacific J. Math. 102 (1982), no. 1, 61–88. MR 682045
  • [16] -, Best simultaneous Diophantine approximations. III. Approximations to a basis of a non-totally real cubic field, in preparation.
  • [17] Serge Lang, Introduction to diophantine approximations, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0209227

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 10F10, 10F20

Retrieve articles in all journals with MSC: 10F10, 10F20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0662052-7
Keywords: Simultaneous Diophantine approximation
Article copyright: © Copyright 1982 American Mathematical Society