Diophantine inequalities with an infinity of solutions

Abstract

Many results in the geometry of numbers assert, in effect, that inequalities of a certain type are soluble in integers, the constant on the right of the inequality being the best possible. Recent work of Mahler often enables one to prove that such an inequality has infinitely many solutions. In this paper we develop the theory of inequalities with infinitely many solutions, and investigate more deeply some of the questions which naturally arise.