Grids with dense values (Q1955668)

Given a continuous function \( F:\mathbb{R}^d \to \mathbb{R} \), a natural question in number theory is to try to analyze the set of values that \(F\) takes on the points of a lattice in \(\mathbb{R}^d\). In this paper, an inhomogeneous variant of this question is discussed: to analyse the set of values that \(F\) takes on grids, i.e., on translated lattices. This discussion is approached from a dynamical point of view which leads to impose some natural assumptions on the function \(F\) under consideration. Let \(X_{d} =G/\Gamma\) where \(G=\mathrm{SL}_{d}(\mathbb{R}), \Gamma=\mathrm{SL}_{d}(\mathbb{Z})\). Let \(\mathbf{H}_{F}=\{g\in G: F\circ g=F\}\), and \(H_{F}= \) the connected component of the identity in \(\mathbf{H}_{F}\). The main result of this paper is: Let \( F:\mathbb{R}^d \to \mathbb{R} \) be nondegenerate and noncompact. Let \(x \in X_{d}\) be a lattice with a nondivergent \(H_{F}\)-orbit. Let \(U< \mathbb{R}^d\) be a subspace, rational with respect to \(x\), and \( w \in \mathbb{R}^d\). Finally, let \(\lambda_{w+U}\) be the translation of the Haar measure supported on the sub-torus \(U+x\) by \(w\). Then, if there exists a divergent sequence \(h_{n}\in H_{F}\) such that \(h_{n}x\) converges and \(h_{n}\) is not almost finite with respect to \(U\), then \(x\) is \(\lambda_{w+U}\)-almost surely grid-\(DV_{F}\). A variety of concrete examples have been given and applications to classical discussions in Diophantine approximations and the geometry of numbers; in particular Minkowski's conjecture regarding products of inhomogeneous forms, have also been given. The main tools to derive the results are the mixing and the coset lemmas.