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Zbl 1024.11051
Bernik, V.I.; Kovalevskaya, E.I.
The distribution of rational points close to a~smooth manifold and Hausdorff dimension. (English)
[J] Acta Math. Inform. Univ. Ostrav. 6, No.1, 31-35 (1998). ISSN 1211-4774

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

MSC 2000:: *11J83 Metric theory of numbers
11J17 Approximation by numbers from a fixed field
11J13 Simultaneous homogeneous approximation, linear forms

Keywords: rational point; Hausdorff dimension

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