On the structure of low-rank matrices that approximate the identity matrix

Consider a matrix $A$ of rank $n$ that approximates the $N\times N$ identity matrix with elementwise error at most $1\mathrm{/}3$. We give a lower bound on the number of elements s.t. $\mathrm{\mid }{A}_{i,j}\mathrm{\mid }>\gamma$, for a certain threshold. Two corollaries are obtained. 1. If $n\le K\log N$ with some $K$, then at least $c(K)N^2$ elements satisfy $\mathrm{\mid }{A}_{i,j}\mathrm{\mid }>c(K)n^{-1\mathrm{/}2}$. This answers a question of B.S. Kashin. 2. The number of nonzero elements in $A$ is at least $c\log (N)\mathrm{/}(n\log (2+n\mathrm{/}\log N))$.