The Discrepancy of a Needle on a Checkerboard

Consider the plane as a checkerboard, with each unit square colored black or white in an arbitrary manner. We show that for any such coloring there are straight line segments, of arbitrarily large length, such that the difference of their white length minus their black length, in absolute value, is at least the square root of their length, up to a multiplicative constant. For the corresponding “finite” problem (N × N checkerboard) we also prove that we can color it in such a way that the above quantity is at most C √ N logN , for any placement of the line segment.