Matrix rounding with low error in small submatrices

We show that any real valued matrix <i>A</i> can be rounded to an integer one <i>B</i> such that the error in all 2 x 2 (geometric) submatrices is less than 1.5, that is, we have |<i>a<inf>ij</inf></i> - <i>b<inf>ij</inf></i>| < 1 and |Σ^<i>i</i>+1<inf><i>k=i</i></inf> Σ^<i>j</i>+1<inf><i>l=j</i></inf>(<i>a</i><inf><i>kl</i></inf> - <i>b</i><inf><i>kl</i></inf>| < 1.5 for all <i>i, j.</i> More precisely, an error of less than 1.5 - 3^-2<i>mn</i> + 3^-d+1 can be achieved in time <i>O</i>(<i>mnd</i>).