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Siegel’s Lemma with additional conditions. (English) Zbl 1192.11018
Summary: Let $$K$$ be a number field, and let $$W$$ be a subspace of $$K^N$$, $$N\geq 1$$. Let $$V_1,\dots,V_M$$ be subspaces of $$K^N$$ of dimension less than dimension of $$W$$. We prove the existence of a point of small height in $$W\backslash \bigcup_{i=1}^m V_i$$, providing an explicit upper bound on the height of such a point in terms of heights of $$W$$ and $$V_1,\dots,V_M$$. Our main tool is a counting estimate we proved for the number of points of a subspace of $$K^N$$ inside of an adelic cube [Monatsh. Math. 147, No. 1, 25–41 (2006; Zbl 1091.11024)].
As corollaries to our main result we derive an explicit bound on the height of a nonvanishing point for a decomposable form and an effective subspace extension lemma.

##### MSC:
 11D04 Linear Diophantine equations 11H06 Lattices and convex bodies (number-theoretic aspects) 11H46 Products of linear forms 11J13 Simultaneous homogeneous approximation, linear forms
##### Keywords:
lattices; linear forms; Diophantine approximation; height
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##### References:
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