The minimum overlap problem revisited

TL;DR

This paper improves the upper bound estimate of the minimum maximum difference overlap in a partition problem by using step functions to model densities, refining previous bounds.

Contribution

It introduces a new step function approach that tightens the upper bound on the limit of M(n)/n in the minimum overlap problem.

Findings

Upper bound improved from 0.382002 to 0.380926

Step functions effectively estimate the limit of M(n)/n

Method provides a new tool for analyzing difference overlaps

Abstract

For a given partition of (1, 2, ..., 2n) into two disjoint subsets A and B with n elements in each, consider the maximum number of times any integer occurs as the difference between an element of A and an element of B. The minimum value of this maximum (over all partitions) is denoted by M(n). By a result of Swinnerton-Dyer, one way to estimate lim M(n)/n from above is to give step functions that describe the density of A, say, throughout the interval [1, 2n] for a large n rather than looking for explicit partitions. A step function that improves the upper bound from 0.382002... to 0.380926... is given.