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An adelic Minkowski-Hlawka theorem and an application to Siegel’s lemma. (English) Zbl 0858.11034
A subset $$C$$ of $$\mathbb{R}^n$$ is said to be a star body if for all $$x\in C$$ and every $$r\in \mathbb{R}$$ with $$|r|\leq 1$$, $$rx$$ is an interior point of $$C$$. The Minkowski-Hlawka theorem states that, if $$C$$ is a star body of volume less than $$2\zeta(n) = 2 \sum_{k = 1}^\infty k^{-n}$$, there exists a lattice $$\Lambda \subset \mathbb{R}^n$$ of determinant 1 having no point in common with $$C$$; stated otherwise, there is a matrix $$A\in\text{SL} (n,\mathbb{R})$$ such that $$A(\mathbb{Z}^n) \cap C= \emptyset$$. The author proves an analogue of this for the adele rings of a number field and of a function field over $$\mathbb{F}_p$$. As a consequence, he shows that in the Bombieri-Vaaler Siegel’s lemma over $$K$$ the dependence on the discriminant of $$K$$ is best possible. Analogously, the author shows that in a version of Siegel’s lemma of his over a function field over $$\mathbb{F}_p$$ [Mich. Math. J. 42, 147-162 (1995; Zbl 0830.11024)], the dependence on the genus is best possible.
We state the author’s generalisation of the Minkowski-Hlawka theorem over a number field $$K$$. Let $$K_A$$ denote the ring of adeles of $$K$$. Let $$\alpha^n$$ be the Haar measure on $$K^n_A$$ normalised such that $$\alpha^n (K^n_A/K^n) =1$$. Choose absolute values $$|\cdot |_v$$ $$(v\in M_K)$$ on $$K$$ satisfying the product formula $$\prod_{v\in M_K} |x |_v=1$$ for $$x\in K^*$$. A measurable subset $$C$$ of $$K^n_A$$ is called a star body if for every $$a\in K^*_A$$ with local components satisfying $$|a_v |_v\leq 1$$ for every $$v\in M_K$$ and for every $$x\in C$$, $$ax$$ is an interior point of $$C$$. The author shows that if $$C$$ is a star body with measure $$\alpha^n(C) < \zeta_K(n) \cdot \{w(K) / n^{r(K)} h(K)R(K)\}$$, then there is a matrix $$A\in \text{GL} (n,K_A)$$ with $$\prod_{v\in M_K} |\text{det} A_v |_v=1$$ (where $$A_v$$ is the local component of $$A$$ at $$v)$$ such that $$A(K^n) \cap C = \emptyset$$. Here, as usual, $$\zeta_K$$ denotes the Dedekind zeta function associated to $$K$$ and $$w(K)$$, $$r(K)$$, $$h(K)$$, $$R(K)$$ the number of roots of unity, the unit rank, the class number and the regulator of $$K$$, respectively.

##### MSC:
 11H16 Nonconvex bodies 11R56 Adèle rings and groups 11R58 Arithmetic theory of algebraic function fields
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