Abstract
For every dimensiond≥1 there exists a constantc=c(d) such that for alln≥1, every set of at leastcn lattice points in thed-dimensional Euclidean space contains a subset of cardinality preciselyn 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.
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Alon, N., Dubiner, M. A lattice point problem and additive number theory. Combinatorica 15, 301–309 (1995). https://doi.org/10.1007/BF01299737
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DOI: https://doi.org/10.1007/BF01299737