A lattice point problem and additive number theory (Q1900180)

Let \(f(n,d)\) denote the minimal number such that every set of \(f(n,d)\) lattice points in a \(d\) dimensional space contains \(n\) points whose centroid (mean) is also a lattice point. Here the estimate \(f(n,d) < (cd \log d)^dn\) is proved, a sharp one for fixed \(d\) and \(n \to \infty\). The proof combines Plünnecke's ideas, expansion properties of graphs with given eigenvalues, and classical exponential sums to evaluate the eigenvalues of certain Cayley graphs.