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Abstract: Consider the plane as a checkerboard, with each unit square colored black orwhite in an arbitrary manner. In a previous paper we showed that for any suchcoloring there are straight line segments, of arbitrarily large length, suchthat the difference of their white length minus their black length, in absolutevalue, is at least the square root of their length, up to a multiplicativeconstant. For the corresponding -finite- problem $N \times N$ checkerboard wehad proved that we can color it in such a way that the above quantity is atmost $C \sqrt{N \log N}$, for any placement of the line segment. In thisfollowup we show that it is possible to color the infinite checkerboard withtwo colors so that for any line segment $I$ the excess of one color overanother is bounded above by $C \epsilon \Abs{I}^{\frac12+\epsilon}$, for any$\epsilon>0$. We also prove lower bounds for the discrepancy of circular arcs.Finally, we make some observations regarding the $L^p$ discrepancies forsegments and arcs, $p<2$, for which our $L^2$-based methods fail to give anyreasonable estimates.



Autor: Alex Iosevich, Mihail N. Kolountzakis

Fuente: https://arxiv.org/







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