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Summary: A lattice point problem and additive number theory
Noga Alon and Moshe Dubiner
Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
Abstract
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 car-
dinality 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.
1 Introduction
Let f(n, d) denote the minimum possible number f so that every set of f lattice points in the
d-dimensional Euclidean space contains a subset of cardinality n whose centroid is also a lattice
point. The problem of determining or estimating f(n, d) was suggested by Harborth [12], and
studied by various authors.
By an old result of Erdos, Ginzburg and Ziv [8], f(n, 1) = 2n-1 for all n. For the general case,
the following simple bounds are proved in [12]:
(n - 1)2d
+ 1 f(n, d) (n - 1)nd
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