Almost periodic functions and an analytical method of solving the number partitioning problem

TL;DR

This paper explores the limit sets of almost periodic functions and introduces an analytical method that directly solves the number partitioning problem, offering an alternative to existing algorithms like Karmarkar--Karp.

Contribution

It establishes a new analytical approach linking almost periodic functions to the partition problem, providing exact formulas for limit sets and solutions.

Findings

The limit set of an almost periodic function is a ring between infimum and supremum values.

The method yields exact solutions to partition problems, outperforming some traditional algorithms.

Examples demonstrate the effectiveness of the analytical approach in specific cases.

Abstract

In the present paper, we study the limit sets of the almost periodic functions $f(x)$. It is interesting that the values $r=\inf|f(x)|$ and $R=\sup|f(x)|$ may be expressed in the exact form. We show that the ring $r\leq |z|\leq R$ is the limit set of the almost periodic function $f(x)$ (under some natural conditions on $f$). The exact expression for $r$ coincides with the well known partition problem formula and gives a new analytical method of solving the corresponding partition problem. Several interesting examples are considered. For instance, in the case of the five numbers, the well-known Karmarkar--Karp algorithm gives the value $m=2$ as the solution of the partition problem in our example, and our method gives the correct answer $m=0.$ The figures presented in Appendix illustrate our results.