Discrepancy of line segments for general lattice checkerboards

TL;DR

This paper extends the study of checkerboard discrepancy to more general lattice-based backgrounds, analyzing how the discrepancy behaves for various shapes beyond squares, using spectral and tiling properties.

Contribution

It generalizes previous discrepancy results from square backgrounds to arbitrary fundamental domains of lattices, employing new proof techniques based on spectral properties.

Findings

Discrepancy of order square root of curve length persists for general lattice backgrounds.

Fourier decay properties of indicator functions are replaced by tiling and spectral arguments.

Results apply to a broader class of lattice shapes beyond squares.

Abstract

In a series of papers recently "checkerboard discrepancy" has been introduced, where a black-and-white checkerboard background induces a coloring on any curve, and thus a discrepancy, i.e., the difference of the length of the curve colored white and the length colored black. Mainly straight lines and circles have been studied and the general situation is that, no matter what the background coloring, there is always a curve in the family studied whose discrepancy is at least of the order of the square root of the length of the curve. In this paper we generalize the shape of the background, keeping the lattice structure. Our background now consists of lattice copies of any bounded fundamental domain of the lattice, and not necessarily of squares, as was the case in the previous papers. As the decay properties of the Fourier Transform of the indicator function of the square were strongly used before, we now have to use a quite different proof, in which the tiling and spectral properties of the fundamental domain play a role.