Circle discrepancy for checkerboard measures

Abstract: 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.