On the best bound of the minimal twisted height of linear subspaces (Q2431751)

This paper expands the results of \textit{I. Aliev, A. Schinzel}, and \textit{W. M. Schmidt} [Monatsh. Math. 144, No. 3, 177--191 (2005; Zbl 1117.11036)] on the estimates of generalized Hermite-Rankin constants for the rational case to the case of global fields and nontrivial elements of the corresponding adele group. Let \(V\) be a vector space of a global field \(k\), \(g\) be an element of the adele group \(\text{GL}(V({\mathbb A}))\) and \(H_g\) the twisted height defined on the \(k\)-subspace of \(V\). The author shows that the square root of the generalized Hermite-Rankin constant for \(k\) gives the best upper bound of the function \(\Gamma_k^{(m,n)}(g)=\sup_X\inf_YH_g(Y)H_g(X)^{-(m/n)}\), where \(X\) runs over all \(m\)-dimensional \(k\)-subspaces of \(V\) and \(Y\) runs over all \(n\)-dimensional \(k\)-subspaces of \(X\).