Values of pairs involving one quadratic and one linear form at S-integral points

TL;DR

This paper establishes the existence of S-integral solutions to certain diophantine inequalities involving quadratic and linear forms, using advanced tools from algebraic groups and homogeneous dynamics.

Contribution

It introduces new conditions under which simultaneous diophantine inequalities with quadratic and linear forms have solutions, employing techniques from algebraic groups and ergodic theory.

Findings

Existence of S-integral solutions under specific conditions

Application of strong approximation in algebraic groups

Use of Ratner's rigidity theorem in diophantine problems

Abstract

We prove the existence of S-integral solutions of simultaneous diophantine inequalities for pairs (Q,L) involving one quadratic form and one linear form satisfying some arithmetico-geometric conditions. The proof uses strong approximation in algebraic groups and Ratner's topological rigidity of unipotent actions on homogeneous spaces.