An Inverse Theorem for Sums of Sets of Lattice Points Original Research Article

Let A be a subset of an n-dimensional Euclidean space V, and let A denote the cardinality of A. The sumset 2A is defined as the set of all vector of the form a1 + a2, where a1, a2 set membership, variant A. Theorem: Let n ≥ 2 and 1 < c < 2n. There exist constants k*0 = k*0(n, c) and ε0 = ε0(n, c) with the property that if A is a finite subset of V such that A ≥ k*0 and 2A ≤ c A, then there exists a hyperplane H in V such that A ∩ H > ε0 A.