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On pairs of cubic diophantine inequalities. (English) Zbl 0759.11009

H. Davenport and H. Heilbronn [J. Lond. Math. Soc. 21, 185-193 (1946; Zbl 0060.119)] proved that if \(Q({\mathbf x})=\sum^ 5_{j=1}\lambda_ jx^ 2_ j\) is an indefinite quadratic form with real coefficients \(\lambda_ j\), such that at least one of the ratios \(\lambda_ i/\lambda_ j\) is irrational, then for any \(\varepsilon>0\) there exist integers, \(x_ 1,\dots,x_ 5\), not all zero, such that \(| Q({\mathbf x})|<\varepsilon\). The method applied in their proof can be extended without difficulties to treat \(\sum^ s_{j=1}\lambda_ jx^ k_ j\) where \(k\geq 2\) and \(s=s(k)\), eg. \(s=2^ k+1\). R. J. Cook [Acta Arith. 25, 337-346 (1974; Zbl 0278.10021)] extended the result to a pair of diagonal quadratic forms. In the present paper, the authors take a step further to obtain a result for a pair of diagonal cubic forms with a better bound than \(\varepsilon\).
Let \(\lambda_ j\), \(\mu_ j\), \(j=1,\dots,15\) be algebraic numbers and put \[ \Lambda({\mathbf x})=\sum^{15}_{j=1}\lambda_ jx^ 3_ j,\;M({\mathbf x})=\sum^{15}_{j=1}\mu_ jx^ 3_ j\quad\text{and}\quad L_{ijk}(\theta)=\text{det}\left(\begin{matrix} \theta_ 1 & \theta_ 2 & \theta_ 3 \\ \lambda_ i & \lambda_ j & \lambda_ k \\ \mu_ i & \mu_ j & \mu_ k\end{matrix} \right). \] If not all the ternary linear forms \(L_{ijk}\) have coefficients linearly dependent over the rationals, then for \(\sigma<3/56\) the simultaneous inequalities \[ |\Lambda({\mathbf x})|<(\max| x_ j|)^{- \sigma}\quad\text{and}\quad| M({\mathbf x})|<\max(| x_ j|)^{-\sigma} \] have infinitely many solutions in integers \(x_ 1,\dots,x_{15}\). – The proof is a modification of Cook’s [ibid].

MSC:

11D75 Diophantine inequalities
11J25 Diophantine inequalities
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References:

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