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On a problem in simultaneous Diophantine approximation: Schmidt’s conjecture. (English) Zbl 1268.11105

Throughout \(\|\cdot\|\) denotes the distance of a real number to the nearest integer. A real number is said badly approximable if there exists a positive constant \(c(x)\) such that \(\|q(x)\|> c(x)q^{-1}\;\;\;\forall q\in \mathbb N\). Let us note \(\mathbf {Bad}\) the set of badly approximable numbers. \(\mathbf {Bad}\) is of Lebesgue measure zero. For a precise definition and details on the Hausdorff dimension \(\dim X\) of a set \(X\) used in this article, see for instance [K. Falconer, Fractal geometry: mathematical foundations and applications. Chichester etc.: John Wiley (1990; Zbl 0689.28003)]. A result of Jarnik states that \(\dim \mathbf {Bad}=1\). The authors use the following generalization of \(\mathbf {Bad}\) to the plane \(\mathbb R^2\): given a pair of real numbers \((i,j)\) such that \[ 0\leq i,j\leq 1\text{\;and\;}i+j=1, \tag{e2} \] a point \((x,y)\in \mathbb R^2\) is said to be \((i,j)\)-badly approximable if there exists a positive constant \(c(x,y)\) such that \(\max\{\|qx\|^{1/i},\|qy\|^{1/j}\}>c(x,y)q^{-1} \;\;\;\forall q\in\mathbb N\). The Schmidt Conjecture (1982) is: For any \((i_1, j_1)\) and \((i_2,j_2)\) satisfying (e2) we have \(\mathbf {Bad}(i_1,j_1)\cap \mathbf {Bad}(i_2,j_2)\not=\emptyset\) [W. M. Schmidt, Approximations diophantiennes et nombres transcendants, Colloq. Luminy/Fr. 1982, Prog. Math. 31, 271–287 (1983; Zbl 0529.10032)]. For further background, (see, H. Davenport [Mathematika, Lond. 11, 50-58 (1964; Zbl 0122.05903)]; D. Kleinbock and B. Weiss [Adv. Math. 223, No. 4, 1276–1298 (2010; Zbl 1213.11148)]; S. Kristensen, R. Thorn and S. Velani [Adv. Math. 203, No. 1, 132–169 (2006; Zbl 1098.11039)]; [A. D. Pollington and S. L. Velani, Acta Math. 185, No. 2, 287–306 (2000; Zbl 0970.11026), J. Lond. Math. Soc., II. Ser. 66, No.1, 29–40 (2002; Zbl 1026.11061)].
The authors investigate the intersection of the sets \(\mathbf {Bad}(i_t,j_t)\) along fixed vertical lines in the \((x,y)\)-plane: let \(L_x\) denote the line parallel to the \(y\)-axis passing through the point \((x,0)\) and, for any real number \(0\leq i\leq 1\), define the set \[ \mathbf {Bad}(i):=\{ x\in \mathbb R : \exists\;c(x) >0\text{\;so that\;} \|qx\|>c(x)q^{-1/i}\;\;\;\forall q\in\mathbb N\}. \] The authors prove: {Theorem} Let \((i_t,j_t)\) be a countable number of pairs of real numbers satisfying ({e2}) and let \(i:=\sup\{i_t : t\in\mathbb N\}\). Suppose that \[ \liminf_{t\to\infty}\min\{i_t,j_t\}>0.\tag{e5} \] Then, for any \(\theta\in \mathbf{Bad}(i)\), we have \(\dim\Big(\bigcap_{t=1}^\infty \mathbf {Bad}(i_t,j_t)\cap L_\theta\Big)=1\).
The hypothesis \(\theta\in \mathbf {Bad}(i)\) is necessary, if not the intersection of the sets \(\mathbf {Bad}(i_t,j_t)\) along the line \(L_\theta\) is empty. As a consequence, the authors prove:
Let \((i_1,j_1),\dots,(i_d,j_d)\) be a finite number of pairs of real numbers satisfying (e2). Then \(\dim \Big(\bigcap_{t=1}^d \mathbf {Bad}(i_t,j_t)\Big)=2\).
This theorem implies that Schmidt’s conjecture is true. The authors prove also the following generalization:
Let \((i_t,j_t)\) be a countable number of pairs of real numbers satisfying (e2). Suppose that (e5) is also satisfied. Then \(\dim\Big(\bigcap_{t=1}^\infty\mathbf {Bad}(i_t,j_t)\Big)=2\).

MSC:

11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
11K60 Diophantine approximation in probabilistic number theory
11J83 Metric theory
11J13 Simultaneous homogeneous approximation, linear forms
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References:

[1] J. W. S. Cassels, An Introduction to the Geometry of Numbers, New York: Springer-Verlag, 1997. · Zbl 0866.11041
[2] H. Davenport, ”A note on Diophantine approximation. II,” Mathematika, vol. 11, pp. 50-58, 1964. · Zbl 0122.05903 · doi:10.1112/S0025579300003478
[3] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Chichester: John Wiley & Sons Ltd., 1990. · Zbl 0689.28003
[4] D. Kleinbock and B. Weiss, ”Modified Schmidt games and Diophantine approximation with weights,” Adv. Math., vol. 223, iss. 4, pp. 1276-1298, 2010. · Zbl 1213.11148 · doi:10.1016/j.aim.2009.09.018
[5] S. Kristensen, R. Thorn, and S. Velani, ”Diophantine approximation and badly approximable sets,” Adv. Math., vol. 203, iss. 1, pp. 132-169, 2006. · Zbl 1098.11039 · doi:10.1016/j.aim.2005.04.005
[6] A. D. Pollington and S. L. Velani, ”On a problem in simultaneous Diophantine approximation: Littlewood’s conjecture,” Acta Math., vol. 185, iss. 2, pp. 287-306, 2000. · Zbl 0970.11026 · doi:10.1007/BF02392812
[7] A. D. Pollington and S. L. Velani, ”On simultaneously badly approximable numbers,” J. London Math. Soc., vol. 66, iss. 1, pp. 29-40, 2002. · Zbl 1026.11061 · doi:10.1112/S0024610702003265
[8] W. M. Schmidt, ”Open problems in Diophantine approximation,” in Diophantine Approximations and Transcendental Numbers, Boston, MA: Birkhäuser, 1983, vol. 31, pp. 271-287. · Zbl 0529.10032
[9] A. Venkatesh, ”The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture,” Bull. Amer. Math. Soc., vol. 45, iss. 1, pp. 117-134, 2008. · Zbl 1194.11075 · doi:10.1090/S0273-0979-07-01194-9
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