The discrepancy of a needle on a checkerboard

TL;DR

This paper investigates the maximum discrepancy in white and black lengths along line segments on an arbitrarily colored checkerboard, establishing bounds for both infinite and finite cases.

Contribution

It proves the existence of arbitrarily large line segments with significant discrepancy and constructs colorings for finite boards minimizing this discrepancy.

Findings

Existence of arbitrarily large segments with discrepancy proportional to the square root of their length.

Construction of checkerboard colorings that limit discrepancy to C√(N log N) for finite N.

Quantitative bounds on discrepancy in infinite and finite checkerboard colorings.

Abstract

Consider the plane as a checkerboard, with each unit square colored black or white in an arbitrary manner. We show 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 also prove 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.