Daniel Berend

A d-dimensional parallelepiped in N is a set of the form {m + ∑_{i ε{lunate} S}m_i: S ⊆ {1, 2,..., d}} for some positive integers m, m₁, m₂,..., 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 2d - 1} + O(N ^{3 4 - 1 2d - 1}). A result of a similar nature is established for parallelepipeds satisfying m₁|m₂| ... | m_d.