The number of submatrices of a given type in a Hadamard matrix and related results

Let a, b be fixed positive integers. It is proved that if every relatively large sub-matrix of a i-l-matrix N contains about the same number of both entries, then all 2 " b a by b matrices occur asymptotically the same number of times as a submatrix of N. A similar statement is shown to hold for graphs with uniformly distributed edges.