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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 [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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