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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.

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