Adelic geometry of numbers and generalized Siegel lemmas
Given a finite dimensional vector space over a number field, a Siegel's Lemma provides an explicit upper bound for a nonzero vector of minimal height. Here the author achieves a similar estimate for a vector which does not belong to a finite union of given proper subspaces. Such questions were already investigated, with a different point of view, by \textit{L.~Fukshansky} [Monatsh. Math. 147, No. 1, 25--41 (2006; Zbl 1091.11024); J. Number Theory 120, No. 1, 13--25 (2006; Zbl 1192.11018)]. The results of the paper under review are stronger. This paper is written in the language of adelic vector bundles; a partial translation in terms of geometry of numbers is given. One of the tools is an adelic variant of a theorem of Henk on the number of lattice points of a centrally convex body in terms of the successive minima of the body [\textit{M. Henk}, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 70, 377--384 (2002; Zbl 1126.11034)].
scientific article; zbMATH DE number 5620123