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.
Citation
Mihail N. Kolountzakis. Ioannis Parissis. "Circle discrepancy for checkerboard measures." Illinois J. Math. 56 (4) 1297 - 1312, Winter 2012. https://doi.org/10.1215/ijm/1399395833
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