Parallelepipeds in sets of integers

AbstractA d-dimensional parallelepiped in N is a set of the form {m + ∑i ϵ Smi: S ⊆ {1, 2,…, d}} for some positive integers m, m1, m2,…, md. It is proved that a subset of {1, 2,…, N} not containing a d-dimensional parallelepiped is of cardinality not exceeding N1 − 12d − 1 + O(N34 − 12d − 1). A result of a similar nature is established for parallelepipeds satisfying m1|m2| … | md