Overview
This research addresses the concept of strongly complete sets in natural numbers, which refers to sets for which every sufficiently large integer can be expressed as a sum of distinct elements, even after the removal of a finite number of elements. The study presents a criterion for a set $A\subseteq\mathbb{N}$ to be strongly complete. This criterion involves conditions on the density of elements within specific intervals and on the distribution of elements with respect to real numbers modulo one.
Research Context
The study builds upon existing mathematical work, specifically referencing previous findings by Bergelson and Simmons. A primary objective within this context is to resolve a conjecture proposed by Erd\H{o}s in 1961. The current investigation also explores connections to several other problems related to completeness, including two problems posed by Erd\H{o}s and Graham.
Approach
The approach involves demonstrating that a set $A\subseteq\mathbb{N}$ satisfies the condition of strong completeness under two specific circumstances. The first condition specifies a minimum density for elements of $A$ within dyadic intervals. The second condition relates to the divergence of a sum involving elements of $A$ and real numbers $\theta$.
Findings
The research identifies a two-part criterion for a set $A\subseteq\mathbb{N}$ to be strongly complete:
- The cardinality of the intersection of $A$ with dyadic intervals $(2^k, 2^{k+1}]$ must be greater than or equal to 6 for every sufficiently large $k \in \mathbb{N}$, i.e., $|A\cap(2^k,2^{k+1}]|\ge6$.
- The sum $\sum_{a\in A}\|a\theta\|$ must diverge to infinity for all $\theta \in \mathbb{R}\setminus\mathbb{Z}$.
This finding is stated to resolve a conjecture put forth by Erd\H{o}s in 1961. The methodology developed in the study also permits the establishment of a more generalized strong-completeness criterion. This generalized criterion applies when suitable ordered blocks are used in place of the specific dyadic intervals $(2^k, 2^{k+1}]$. The study further discusses various applications of these results and their interrelations with other problems concerning completeness, including those raised by Erd\H{o}s and Graham.
Why This Matters
The resolution of Erd\H{o}s's 1961 conjecture advances the understanding of strongly complete sets in number theory. The established strong-completeness criterion provides specific conditions under which these sets can be identified. The generalized criterion, accommodating ordered blocks beyond dyadic intervals, indicates broader applicability of the underlying principles.