Matrix approximation and Tusnady's problem

We consider the problem of approximating a given matrix by ail integer one such that in all geometric submatrices the sum of the entries does not change by much. We show that for all integers m, n >= 2 and real matrices A epsilon R-mxn there is an integer matrix B epsilon Z(mxn) such that broken vertical bar Sigma(i epsilon I) Sigma(j epsilon J) (a(ij)-b(ij))broken vertical bar < 4 log (min{m,n}) holds for all intervals I subset of vertical bar n vertical bar. Such a matrix can be Computed in time 0(mn log (min (m, n})). The result remains true if we add the requirement vertical bar a(ij) - b(ij)vertical bar < 2 for all i epsilon [m], j epsilon [n]. This is surprising, as the slightly stronger requirement vertical bar a(ij) - b(ij)vertical bar < I makes the problem equivalent to Tusnady's problem. (c) 2005 Elsevier Ltd. All rights reserved.