Discrepancy One among Homogeneous Arithmetic Progressions

TL;DR

This paper studies the discrepancy of finite sets of homogeneous arithmetic progressions, providing exact results for small sets and proving NP-hardness for the general case.

Contribution

It characterizes when the discrepancy is exactly 1 for sets of size four or less and establishes NP-hardness for the general problem.

Findings

Discrepancy equals 1 for sets of size ≤ 4 is characterized.

NP-hardness of deciding discrepancy 1 for arbitrary sets.

Provides complexity results related to Erdos' problem.

Abstract

We investigate a restriction of Paul Erdos' well-known problem from 1936 on the discrepancy of homogeneous arithmetic progressions. We restrict our attention to a finite set S of homogeneous arithmetic progressions, and ask when the discrepancy with respect to this set is exactly 1. We answer this question when S has size four or less, and prove that the problem for general S is NP-hard, even for discrepancy 1.