CIRCLE DISCREPANCY FOR CHECKERBOARD MEASURES

摘要

Consider the plane as a union of congruent unit squares in a checkerboard pattern, each square colored black or white in an arbitrary manner. The discrepancy of a curve with respect to a given coloring is the difference of its white length minus its black length, in absolute value. We show that for every radius t ≥ 1 there exists a full circle of radius either t or 2t with discrepancy greater than ct~(1/2) for some numerical constant c ＞ 0. We also show that for every t ≥ 1 there exists a circular arc of radius exactly t with discrepancy greater than ct~(1/2). Finally, we investigate the corresponding problem for more general curves and their interiors. These results answer questions posed by Kolountzakis and Iosevich.