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arXiv · 1201.5544

Circle discrepancy for checkerboard measures

Mihail N. Kolountzakis·Ioannis Parissis
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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.

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Mihail N. Kolountzakis, Ioannis Parissis. 2012-01-26. Circle discrepancy for checkerboard measures. https://arxiv.org/abs/1201.5544

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