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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. R. Alexander, J. Beck, and W. W. L. Chen, Geometric discrepancy theory and uniform distribution, in: Handbook of discrete and computational geometry, CRC Press Ser. Discrete Math. Appl., CRC (Boca Raton, FL, 1997), pp. 185–207.
J. Beck and W. W. L. Chen, Irregularities of distribution, Cambridge Tracts in Mathematics, 89, Cambridge University Press (Cambridge, 2008).
B. Chazelle, The Discrepancy Method, Cambridge University Press (Cambridge, 2000).
M. Drmota and R. F. Tichy, Sequences, Discrepancies and Applications, Lecture Notes in Mathematics, 1651, Springer-Verlag (Berlin, 1997).
A. Iosevich and M. N. Kolountzakis, The discrepancy of a needle on a checkerboard. II, Unif. Distrib. Theory, 5 (2010), 1–13.
M. N. Kolountzakis, The study of translational tiling with Fourier Analysis, in: Fourier Analysis and Convexity, Birkhäuser (Boston, 2004), pp. 131–187.
M. N. Kolountzakis, The discrepancy of a needle on a checkerboard, Online J. Anal. Comb., 3 (2008), Art. 7, 5
M. N. Kolountzakis and I. Parissis, Circle discrepancy for checkerboard measures, Illinois J. Math., 56 (2012), 1297–1312.
J. Matousek, Geometric Discrepancy. An illustrated Guide, Algorithms and Combinatorics, 18, Springer-Verlag (Berlin, 1999).
A. D. Rogers, Lower bounds on strip discrepancy for nonatomic colorings, Monatsh. Math., 130 (2000), 311–328.
A. D. Rogers, A functional from geometry with applications to discrepancy estimates and the Radon transform, Trans. Amer. Math. Soc., 341 (1994), 275–313.
A. D. Rogers, Irregularities of distribution with respect to strips, Acta Math. Hungar., 110 (2006), 13–21.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author has been partially supported by the “Aristeia II” action (Project FOURIERDIG) of the operational program Education and Lifelong Learning and is co-funded by the European Social Fund and Greek national resources.
Rights and permissions
About this article
Cite this article
Kolountzakis, M.N. Discrepancy of line segments for general lattice checkerboards. Anal Math 42, 31–41 (2016). https://doi.org/10.1007/s10476-016-0103-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10476-016-0103-3