## 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
Full Text:

### 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 [2] Sprindžuk V.: Metric theorem of Diophantine approximation. 1979. Wiley, New York ( [3] Kovalevskaya E.: Many-dimensional extremal surface of Sprindžuk and Hausdorff dimension. Vesci Acad. Nauk Belarus. 1996, N 3, pp.H4- 116 [4] Bernik V., Melnichuk Yu.: Diophantine approximation and Hausdorff dimension. 1988. Minsk [5] Cassels J. W. S.: Introduction to Diophantine approximation. 1957. Cambridge. · Zbl 0098.26301 [6] Bernik V.: On influence of Hausdorff dimension upon the distribution of rational points close to smooth curves. 1997. Collection of papers in honour in Prof. V. Sprindžuk (1936-1987). Minsk. Institute of Math. Acad. Sci. Belarus, pp.5-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.