Approximating sparse binary matrices in the cut-norm

The cut-norm ‖A‖C of a real matrix A=(aij)i∈R,j∈S is the maximum, over all I⊂R, J⊂S of the quantity |∑i∈I,j∈Jaij|. We show that there is an absolute positive constant c so that if A is the n by n identity matrix and B is a real n by n matrix satisfying ‖A−B‖C≤116‖A‖C, then rank(B)≥cn. Extensions to denser binary matrices are considered as well.