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The distribution of rational points close to a smooth manifold and Hausdorff dimension. (English) Zbl 1024.11051

Let \(X\subseteq \mathbb R^m\) be a domain. For \(1\leq j\leq m\) and \((x_1, \dots ,x_{j-1},x_{j+1},\dots ,x_m)\) \(\in \mathbb R^{m-1}\) let \(X_j(x_1,\dots ,x_{j-1},x_ {j+1},\dots ,x_m)\) be the set of all \(x_j\) such that \((x_1,\dots ,x_m)\in X.\) For \(1\leq j\leq n\) let \(f_j:X\rightarrow \mathbb R\) be three times continuously differentiable and \(\det (\partial ^2f_j/\partial x_1\partial x_k)_{1\leq j,k\leq n}\neq 0\) almost everywhere in \(X.\) Assume further that there is a positive constant \(K\) such that for all \(c\in \mathbb Z^n,\) all \(j,\) \(1\leq j\leq m\) and all \((x_1,\dots ,x_{j-1},x_{j+1},\dots ,x_m)\) \(\in \mathbb R^{m-1}\) the function \(\varphi :X_j(x_1,\dots ,x_{j-1},x_{j+1},\dots ,x_m)\rightarrow \mathbb R,\) \(\varphi (x_ j)=\sum \limits _{i=1}^nc_i\frac {\partial ^2f_i}{\partial x_1\partial x_j}(x)\) is piecewise monotone with at most \(K\) pieces. For \(v>(m+n)^{-1}\) let \(M (v)\) be the set of all \(x\in X\) such that \(\max \limits _{1\leq i\leq m, 1\leq j\leq n}(\|x_iq \|,\|f_j(x)q \|)<q^{-v}\) has infinitely many solutions \(q\in \mathbb N.\) The authors prove that for \(m>n^2-n+1\) the Hausdorff dimension \(\dim M(v)\) satisfies \(\dim M(v)\geq \frac {m-vn}{v+1}.\)
The method of proof uses the regular systems constructed by A. Baker and W. M. Schmidt.

MSC:

11J83 Metric theory
11J17 Approximation by numbers from a fixed field
11J13 Simultaneous homogeneous approximation, linear forms
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References:

[1] Baker A., Schmidt W. M.: Diophantine approximation and Hausdorff dimension. Proc. London Math. Soc. 1970. Vol. 21, part 1, pp. 1-11. · Zbl 0206.05801
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