On the structures of subset sums in higher dimension

TL;DR

This paper investigates the structure of subset sums in higher dimensions, revealing the presence of dense sets and generalized arithmetic progressions, which enhances understanding of their complexity and structure.

Contribution

It introduces a detailed analysis of the structure of subset sums in higher dimensions, highlighting the occurrence of dense sets and generalized arithmetic progressions.

Findings

Subset sums can form dense sets in higher dimensions

Generalized arithmetic progressions are present in subset sums

Provides structural insights relevant to the knapsack problem

Abstract

A given subset $A$ of natural numbers is said to be complete if every element of $\N$ is the sum of distinct terms taken from $A$. This topic is strongly connected to the knapsack problem which is known to be NP complete. The main goal of the paper is to study the structure of subset sums in a higher dimension. We show 'dense' sets and generalized arithmetic progrssions in subset sums of certain sets.