### abstract

- A d -dimensional parallelepiped in N is a set of the form { m + ∑ i ϵ S m i : S ⊆ {1, 2,…, d }} for some positive integers m , m 1 , m 2 ,…, m d . It is proved that a subset of {1, 2,…, N } not containing a d -dimensional parallelepiped is of cardinality not exceeding N 1 − 1 2 d − 1 + O(N 3 4 − 1 2 d − 1 ) . A result of a similar nature is established for parallelepipeds satisfying m 1 | m 2 | … | m d .