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An important problem in analytic and geometric combinatorics is estimatingthe number of lattice points in a compact convex set in a Euclidean space. Suchestimates have numerous applications throughout mathematics. In this note, weexhibit applications of a particular estimate of this sort to several countingproblems in number theory: counting integral points and units of bounded heightover number fields, counting points of bounded height over positive definitequaternion algebras, and counting points of bounded height with a fixed supportover global function fields. Our arguments use a collection of heightcomparison inequalities for heights over a number field and over a quaternionalgebra. We also show how these inequalities can be used to obtain existenceresults for points of bounded height over a quaternion algebra, whichconstitute non-commutative analogues of variations of the classical Siegel'slemma and Cassels' theorem on small zeros of quadratic forms.

Lattice point counting and height bounds over number fields and quaternion algebras