For every dimension d≥1 there exists a constant c=c(d) such that for all n≥1, every set of at least cn lattice points in the d-dimensional Euclidean space contains a subset of cardinality precisely n whose centroid is also a lattice point. The proof combines techniques from additive number theory with results about the expansion properties of Cayley graphs with given eigenvalues.
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics
- Computational Mathematics
- Mathematics Subject Classification (1991): 11B75