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Going-up theorems for simultaneous Diophantine approximation. (English) Zbl 1472.11206

Let \(N\geq 1\) be an integer and \(\underline\zeta =(\zeta_1,\dots ,\zeta_N)\in\mathbb R^N\). Let \(\lambda_N(\underline\zeta)\) be the supremum of real \(\lambda\) such that \[ 1\leq x\leq X \quad \textrm{and} \quad \max_{1\leq j\leq N}\mid \zeta_jx-y_j\mid\leq X^{-\lambda} \tag{1}\] has a solution \((x,y_1,\dots ,y_N)\in\mathbb Z^{N+1}\) for all large values of \(X\). Let \(\omega_N(\underline\zeta)\) be the supremum of real \(\omega\) such that \[\max_{1\leq j\leq N}\mid x_j\mid\leq X \quad \textrm{and} \mid x_0+x_1\zeta_1+\dots +x_N\zeta_N\mid\leq X^\omega\tag{2} \] has a solution \((x_0,\dots ,x_N)\in\mathbb Z^{N+1}\) for all large values of \(X\). Let uniformly \(\widehat\lambda_N(\underline\zeta)\) and \(\widehat\omega_N(\underline\zeta)\) respectively be given as the respective suprema such that (1) and (2) have a solution for all large values of \(X\). Let \(\zeta\) be a real number. Then set \(\lambda_N(\zeta)=\lambda_N(\underline\zeta)\) where \(\underline\zeta =(\zeta,\zeta^2,\dots ,\zeta^N)\). Similarly, we define \(\widehat\lambda_N(\zeta)\), \(\omega_N(\zeta)\) and \(\widehat\omega_N(\zeta)\). Then the author proves that
if \(k\geq n\geq 1\) then \[\lambda_k(\zeta)\geq \frac{(n-1)\lambda_n(\zeta)+(k-n)\widehat\lambda_n(\zeta)+n-k} {(n-k)\widehat\lambda_n(\zeta)+k-1}\,, \] \[\lambda_k(\zeta)\geq \frac{(n-1)\lambda_n(\zeta)+(k-1)\widehat\lambda_n(\zeta)+n-k} {(n-1)\lambda_n(\zeta)-(k-1)\widehat\lambda_n(\zeta)+n+k-2}\,, \]
if \(k\geq n\geq 2\) then \[\lambda_k(\zeta)\geq \frac{\omega_n(\zeta)\widehat\omega_n(\zeta)-\omega_n(\zeta)+(n-k)\widehat\omega_n(\zeta)} {(n-2)\omega_n(\zeta)\widehat\omega_n(\zeta)+\omega_n(\zeta)+(k-1)\widehat\omega_n(\zeta)}\,, \]
if \(k\geq n\geq 1\) and \(\underline\zeta_N=(\zeta_1,\dots ,\zeta_N)\) for all \(N=1,\dots ,k\) where \(1,\zeta_1,\dots ,\zeta_k\) are linearly independent over \(\mathbb Q\) then \[\lambda_k(\underline\zeta_k)\geq \frac{(n-1)\lambda_n(\underline\zeta_n)+\widehat\lambda_n(\underline\zeta_n)+n-2} {(k-1)(n-1)\lambda_n(\underline\zeta_n)-\widehat\lambda_n(\underline\zeta_n)+kn-n-k+2}\,. \] In some special cases the author improves these inequalities.

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
11J83 Metric theory
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References:

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