The discrepancy of a needle on a checkerboard, II

TL;DR

This paper explores the discrepancy of colored checkerboards and line segments, establishing bounds for the difference in color lengths and extending results to infinite grids and circular arcs, with implications for discrepancy theory.

Contribution

It demonstrates the possibility of coloring an infinite checkerboard to bound discrepancies for all line segments and provides new bounds for circular arc discrepancies.

Findings

Bounded discrepancy for infinite checkerboard coloring

Lower bounds for circular arc discrepancy

Limitations of $L^p$ discrepancy estimates for p<2

Abstract

Consider the plane as a checkerboard, with each unit square colored black or white in an arbitrary manner. In a previous paper we showed that for any such coloring there are straight line segments, of arbitrarily large length, such that the difference of their white length minus their black length, in absolute value, is at least the square root of their length, up to a multiplicative constant. For the corresponding "finite" problem ($N \times N$ checkerboard) we had proved that we can color it in such a way that the above quantity is at most $C \sqrt{N \log N}$, for any placement of the line segment. In this followup we show that it is possible to color the infinite checkerboard with two colors so that for any line segment $I$ the excess of one color over another 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 for segments and arcs, $p<2$, for which our $L^2$-based methods fail to give any reasonable estimates.