Abstract The classical Khintchine–Groshev theorem is a generalization of Khintchine’s on simultaneous Diophantine approximation, from approximation points in ${\mathbb {R}}^m$ to systems linear forms {R}}^{nm}$. In this paper, we present an inhomogeneous version the that does not carry monotonicity assumption when $nm>2$. Our results bring theory almost line with homogeneous theory, wher...

We describe and generalize S.G. Dani’s correspondence between bounded orbits in the space of lattices and systems of linear forms with certain Diophantine properties. The solution to Margulis’ Bounded Orbit Conjecture is used to generalize W. Schmidt’s theorem on abundance of badly approximable systems of linear forms.

In this paper we find all solutions of four kinds of the Diophantine equations begin{equation*} ~x^{2}pm V_{t}xy-y^{2}pm x=0text{ and}~x^{2}pm V_{t}xy-y^{2}pm y=0, end{equation*}% for an odd number $t$, and, begin{equation*} ~x^{2}pm V_{t}xy+y^{2}-x=0text{ and}text{ }x^{2}pm V_{t}xy+y^{2}-y=0, end{equation*}% for an even number $t$, where $V_{n}$ is a generalized Lucas number. This pape...

‎in this paper‎, ‎we obtain the dimensions of symmetry classes of polynomials associated with‎ ‎the irreducible characters of the dihedral group as a subgroup of‎ ‎the full symmetric group‎. ‎then we discuss the existence of o-basis‎ ‎of these classes‎.

In this paper, we present an algorithm for solving directly linear Dio-phantine systems of both equations and inequations. Here directly means without adding slack variables for encoding inequalities as equalities. This algorithm is an extension of the algorithm due to Contejean and Devie 10] for solving linear Diophantine systems of equations, which is itself a generalization of the algorithm ...