Blichfeldt's theorem

Blichfeldt's theorem is a mathematical theorem in the geometry of numbers, stating that whenever a bounded set in the Euclidean plane has area ${\displaystyle A}$, it can be translated so that it includes at least ${\displaystyle \lceil A\rceil }$ points of the integer lattice. Equivalently, every bounded set of area ${\displaystyle A}$ contains a set of ${\displaystyle \lceil A\rceil }$ points whose coordinates all differ by integers.

This theorem can be generalized to other lattices and to higher dimensions, and can be interpreted as a continuous version of the pigeonhole principle. It is named after Danish-American mathematician Hans Frederick Blichfeldt, who published it in 1914. Some sources call it Blichfeldt's principle or Blichfeldt's lemma.