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Diophantine approximation on affine hyperplanes. (English) Zbl 1223.11092

Throughout \(n\geq 2\) is an integer and \({\mathbf s}=(s_1,\dots,s_n)\in(0,1)^n\) satisfies \(s_1+\dots+s_n=1\). The following two functions are defined for \({\mathbf x}=(x_1,\dots,x_n)\in\mathbb{R}^n\): \[ \|{\mathbf x}\|_{\mathbf s}=\max_{1\leq i\leq n}|x_i|^{1/s_i} \qquad\text{and}\qquad \Pi_+({\mathbf x})=\prod_{i=1}^n\max\{1,|x_i|\}. \] Finally, given \(\psi:\mathbb{N}\to(0,+\infty)\), let \({\mathcal L}_n(\psi,{\mathbf s})\) denote the set of \({\mathbf y}\in\mathbb{R}^n\) such that \[ |{\mathbf q}\cdot{\mathbf y}+p|<\psi(\|{\mathbf q}\|_{\mathbf s}^n) \] holds for infinitely many \((p,{\mathbf q})\in\mathbb{Z}\times(\mathbb{Z}^n\setminus\{0\})\). Here the ‘dot’ denotes the standard inner product. Similarly, let \({\mathcal L}_n(\psi,\times)\) denote the set of \({\mathbf y}\in\mathbb{R}^n\) such that \[ |{\mathbf q}\cdot{\mathbf y}+p|<\psi(\Pi_+({\mathbf q})) \] holds for infinitely many \((p,{\mathbf q})\in\mathbb{Z}\times(\mathbb{Z}^n\setminus\{0\})\). The sets \({\mathcal L}_n(\psi,{\mathbf s})\) and \({\mathcal L}_n(\psi,\times)\) naturally appear in the metric theory of Diophantine approximation, where the measure theoretic properties of Diophantine sets are studied. In particular, when \({\mathbf s}=(1/n,\dots,1/n)\), the set \({\mathcal L}_n(\psi,{\mathbf s})\) was studied by Groshev in 1938, yet the case \(n=1\) was dealt with by Khintchine by his well known result of 1924. To date, the measure theoretic properties of \({\mathcal L}_n(\psi,{\mathbf s})\) and \({\mathcal L}_n(\psi,\times)\) as well as other natural counterparts are fairly completely understood. The more recent trend of metric Diophantine approximation has been to discover the structure of such sets restricted to a proper submanifold of \(\mathbb{R}^n\), or more ‘exotically’ the support of a measure. The seemingly simplest yet interesting case is when the manifold is an affine subspace of \(\mathbb{R}^n\). It is this case that is addressed by the paper under review. To be precise, the paper considers Diophantine approximation on hyperplanes \({\mathcal H}\) in \(\mathbb{R}^n\) given by \[ \textstyle {\mathcal H}=\left\{\Big(x_1,\dots,x_{n-1},a_0+\sum_{i=1}^{n-1}a_ix_i\Big):(x_1,\dots,x_{n-1})\in\mathbb{R}^{n-1}\right\}, \] where \({\mathbf a}=(a_0,\dots,a_{n-1})\in\mathbb{R}^n\) is fixed. The following results are established.
Theorem 1: Let \({\mathbf a}=(a_0,\dots,a_{n-1})\in\mathbb{R}^n\) and there exist \(u<n\) such that \[ \max_i|p_i+a_iq|\geq |q|^{-u} \] for all but finitely many \((q,p_0,\dots,p_{n-1})\in\mathbb{Z}^{n+1}\). Let \({\mathbf s}=(s_1,\dots,s_n)\) be an \(n\)-tuple of positive numbers such that \(s_1+\dots+s_n=1\). Let \(\psi:\mathbb{N}\to(0,+\infty)\) be monotonic and \[ \sum_{q=1}^\infty\psi(q)<\infty. \] Let \({\mathcal H}\) be as above. Then \({\mathcal H}\cap{\mathcal L}_n(\psi,{\mathbf s})\) is of (induced Lebesgue) measure zero on \({\mathcal H}\).
Theorem 2: Let \({\mathcal H}\) and \({\mathbf a}\) be the same as in Theorem 1 and in addition \(a_i\not=0\) for all \(i=1,\dots,n-1\). Let \(\psi:\mathbb{N}\to(0,+\infty)\) be monotonic and \[ \sum_{q=1}^\infty\psi(q)\log^{n-1}\!q<\infty. \] Then \({\mathcal H}\cap{\mathcal L}_n(\psi,{\mathbf s})\) is of (induced Lebesgue) measure zero on \({\mathcal H}\).

MSC:

11J83 Metric theory
11K60 Diophantine approximation in probabilistic number theory
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