If is a full lattice and a measurable convex centrally symmetric set has volume greater than , then contains a nonzero lattice point.
For a real quadratic field with embeddings , the imageis a lattice in of covolume , and the product of the two coordinates is the field norm.
Let for square-free with , so . An integral ideal is a lattice in of covolume . Applying the Minkowski convex body theorem to a disk gives a nonzero satisfying
Articles by others on the same topic
Minkowski's theorem is a fundamental result in the field of number theory and geometry, particularly in the context of convex geometry and the geometry of numbers. The theorem addresses the existence of certain lattice points within convex bodies in Euclidean space.