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TIGHTER BOUNDS FOR THE DISCREPANCY OF BOXES AND POLYTOPES

Part of: Harmonic analysis in several variables Polytopes and polyhedra Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  29 November 2017

Aleksandar Nikolov
Aleksandar Nikolov*
Affiliation:
Department of Computer Science, University of Toronto, Toronto, Canada email anikolov@cs.toronto.edu
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Abstract

Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced. More precisely, we are given an $n$-point set $P$, and a collection ${\mathcal{F}}=\{F_{1},\ldots ,F_{m}\}$ of subsets of $P$, and our goal is color $P$ with two colors, red and blue, so that the maximum over the $F_{i}$ of the absolute difference between the number of red elements and the number of blue elements (the discrepancy) is minimized. Combinatorial discrepancy has many applications in mathematics and computer science, including constructions of uniformly distributed point sets, and lower bounds for data structures and private data analysis algorithms. We investigate the combinatorial discrepancy of geometrically defined systems, in which $P$ is an $n$-point set in $d$-dimensional space, and ${\mathcal{F}}$ is the collection of subsets of $P$ induced by dilations and translations of a fixed convex polytope $B$. Such set systems include systems of sets induced by axis-aligned boxes, whose discrepancy is the subject of the well-known Tusnády problem. We prove new discrepancy upper and lower bounds for such set systems by extending the approach based on factorization norms previously used by the author, Matoušek, and Talwar. We also outline applications of our results to geometric discrepancy, data structure lower bounds, and differential privacy.

MSC classification

Primary: 11K38: Irregularities of distribution, discrepancy
Secondary: 52B11: $n$-dimensional polytopes 42B05: Fourier series and coefficients
Type
Research Article
Information
Mathematika , Volume 63 , Issue 3 , 2017 , pp. 1091 - 1113
Copyright
Copyright © University College London 2017 

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