Blichfeldts LemmaEdit
Blichfeldts lemma is a foundational result in the geometry of numbers. It formalizes a simple, intuitive idea: if you have a lattice in Euclidean space and you look at a region with finite volume, you can translate that region so that it captures a predictable minimum number of lattice points. The lemma is a workhorse in proofs and constructions, providing a bridge between continuous volume and discrete lattice points. It is named after the Danish mathematician Eigil Blichfeldt, who developed the idea in the early 20th century as part of the developing theory of lattices and their distribution in space. For readers new to the topic, the lemma sits alongside core results like Minkowski's theorem and is a staple in the broader project of the geometry of numbers.
In modern language, Blichfeldts lemma asserts a translation principle: given a lattice Λ ⊂ R^n with covolume det(Λ) and a measurable set A ⊂ R^n of finite measure vol(A), there exists a translation x ∈ R^n such that the translated set A+x contains at least floor(vol(A)/det(Λ)) points of Λ. Equivalently, if one averages the number of lattice points in A+x over all translations x modulo a fundamental domain of Λ, the average equals vol(A)/det(Λ); hence some translation achieves at least that bound. This translation-based viewpoint is often framed as a pigeonhole-type argument in a compact space, and it underscores how continuous volume controls discrete lattice membership.
Statement and intuition
- Let Λ ⊂ R^n be a lattice with covolume det(Λ), and let A ⊂ R^n be a measurable set with finite measure vol(A).
- Then there exists x ∈ R^n such that the number of lattice points of Λ in A+x satisfies #(A ∩ (x+Λ)) ≥ floor(vol(A)/det(Λ)).
- A quick corollary: if vol(A) > det(Λ), there exists a translation x for which A+x contains at least one lattice point; if vol(A) > 2 det(Λ), there is a translation with at least two lattice points, and so on.
The proof rests on a simple averaging argument. Projecting A into the torus R^n/Λ and counting lattice points in translates corresponds to examining how often points of Λ land inside A+x as x runs through a fundamental domain. Since the average number of lattice points in A+x over all translations is vol(A)/det(Λ), some translation must realize at least that many lattice points. The method is a clean instance of the pigeonhole principle in a continuous setting.
Historical background
Blichfeldts lemma is associated with Eigil Blichfeldt, who contributed to the early development of the geometry of numbers, a program that seeks to relate the geometry of space to the distribution of integer lattice points. This line of work was contemporaneous with and complemented the foundational results of Minkowski's theorem and the broader study of lattices in geometry of numbers. The lemma remains a standard tool in the field, taught in courses on lattice theory and Diophantine approximation and cited in many modern expositions of the subject. For biographical context, see Eigil Blichfeldt.
Proof outline
A concise route to the lemma uses a measure-theoretic averaging argument over a fundamental domain F of Λ. Consider the function f(x) = #(A ∩ (x+Λ)) as a function on the torus R^n/Λ. Integrating f over F yields an value equal to vol(A). Since F has volume det(Λ), the average value of f over the torus is vol(A)/det(Λ). Therefore there exists some x ∈ F with f(x) ≥ floor(vol(A)/det(Λ)), which yields the claimed translation. The core ideas are the translation invariance of the lattice and a straightforward application of the pigeonhole principle in a compact setting.
Applications and connections
- Minkowski's convex body theorem: Blichfeldts lemma provides a quick route to the existence of a nonzero lattice point inside a sufficiently large symmetric convex body. By translating and recombining copies of a convex body, one can guarantee two points in A whose difference lies in Λ, yielding a nonzero lattice point in the body.
- Diophantine approximation and lattice point counting: the lemma underpins several existence results about how well real vectors can be approximated by rational vectors, by translating regions in space to capture lattice structure.
- Generalizations to other groups and settings: since its original formulation, versions of Blichfeldts idea have been extended to lattices in Lie groups, locally compact groups, and more general measure-theoretic contexts, illustrating the broad reach of the translate-and-count method.
- Educational role: as a relatively short and conceptually transparent argument, the lemma serves as an accessible entry point into the geometry of numbers and as a stepping stone to more advanced results such as Hermite constant bounds and lattice packing problems.