## 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

### Keywords:

rational point; Hausdorff dimension
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### References:

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