Matrix approximation and Tusnády's problem

We consider the problem of approximating a given matrix by an 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 [email protected]?R^m^x^n there is an integer matrix [email protected]?Z^m^x^n such that |@[email protected][email protected][email protected]?J(a"i"j-b"i"j)|<4log"2(min{m,n}) holds for all intervals [email protected]?[m], [email protected]?[n]. Such a matrix can be computed in time O(mnlog(min{m,n})). The result remains true if we add the requirement |a"i"j-b"i"j|<2 for all [email protected]?[m],[email protected]?[n]. This is surprising, as the slightly stronger requirement |a"i"j-b"i"j|<1 makes the problem equivalent to Tusnady's problem.

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