Thue's lemma

Thue's lemma guarantees a nonzero integer point satisfying specified linear inequalities when their volume is sufficiently large. It is a basic existence result in the geometry of numbers, used in Diophantine approximation.

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Lattice: A discrete subgroup of Euclidean space generated by linearly independent vectors. The integer solutions sought by the lemma form a lattice in the relevant space.
Diophantine approximation: The study of approximating real numbers by rational numbers and of bounds on approximation error. Thue's lemma supplies small integer vectors used to derive rational approximations.
Axel Thue: A Norwegian mathematician known for work in number theory, including Diophantine equations and approximation. The lemma bears his name and belongs to the tradition of his Diophantine work.
Blichfeldt's theorem: A theorem asserting that a measurable set of volume greater than a lattice's covolume contains two points differing by a lattice vector. It gives a related volume-to-lattice-point principle through a different formulation.
Convex body: A compact convex subset of Euclidean space with nonempty interior. Convex regions are a standard setting for geometry-of-numbers existence arguments.
Dirichlet's approximation theorem: A theorem ensuring rational approximations to any real number with error bounded in terms of the denominator. It is a central approximation result proved by elementary geometry-of-numbers methods.
Carl Friedrich Gauss: A German mathematician whose work shaped number theory, geometry, and several other fields. Earlier lattice and number-theoretic ideas form part of the background to Thue's methods.
Minkowski's convex body theorem: A theorem guaranteeing a nonzero lattice point in a sufficiently large, centrally symmetric convex body. Its convexity and symmetry hypotheses distinguish it from linear-inequality versions of Thue's lemma.
Determinant of a lattice: The volume of a fundamental region of a lattice. Comparing region volume with lattice covolume quantifies when integer points must occur.
Simultaneous Diophantine approximation: The approximation of several real numbers by fractions sharing a common denominator. Small integer solutions to multiple linear constraints yield simultaneous approximations.