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 $t\geq1$ there exists a full circle of radius either $t$ or $2t$ with discrepancy greater than $c\sqrt{t}$ for some numerical constant $c>0$. We also show that for every $t\geq1$ there exists a circular arc of radius exactly $t$ with discrepancy greater than $c\sqrt{t}$. Finally, we investigate the corresponding problem for more general curves and their interiors. These results answer questions posed by Kolountzakis and Iosevich.