Circle discrepancy for checkerboard measures

TL;DR

The paper proves that in a checkerboard-colored plane, there exist circles and arcs of certain radii with discrepancy growing at least as the square root of the radius, addressing questions in geometric discrepancy theory.

Contribution

It establishes the existence of circles and arcs with large discrepancy in checkerboard patterns, extending understanding of geometric discrepancy for various curves.

Findings

Existence of circles of radius t or 2t with discrepancy > c t^{1/2}

Existence of arcs of radius t with discrepancy > c t^{1/2}

Results answer open questions by Kolountzakis and Iosevich

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.