Block Partitions of Sequences

TL;DR

This paper proves that for sequences of real numbers between 0 and 1, it is possible to partition the sequence into a fixed number of blocks with nearly equal sums, and extends this result in various ways.

Contribution

The paper introduces a new partitioning theorem for sequences with bounded elements, ensuring nearly equal block sums with controlled differences, and explores multiple extensions.

Findings

Existence of partitions with bounded sum differences for sequences in [0,1]

Extension of the partitioning result to broader classes of sequences

Generalization of the equal-sum partitioning concept

Abstract

Given a sequence A=(a1,...,an) of real numbers, a block B of the A is either a set B={ai,...,aj} where i<=j or the empty set. The size b of a block B is the sum of its elements. We show that when 0<=ai<=1 and k is a positive integer, there is a partition of A into k blocks B1,...,Bk with |bi-bj|<=1 for every i, j. We extend this result in many directions.