On the critical values of Burr's problem

TL;DR

This paper investigates the critical values in Burr's problem, determining the thresholds where certain sumset representations of integer sequences change, extending known results for specific cases to general values.

Contribution

The paper generalizes the determination of critical values in Burr's problem from the case of $b_3$ to the broader case of $b_k$, providing new theoretical insights.

Findings

Critical value of $b_3$ is $u+v+1$ for given parameters.

Established the critical value of $b_k$ for general $k$.

Extended previous results to a wider class of sequences.

Abstract

Let $A$ be a sequence of positive integers and $P(A)$ be the set of all integers which are the finite sum of distinct terms of $A$. For given positive integers $u\in\{4,7,8\}\cup\{u:u\ge11\}$ and $v\ge 3u+5$ we know that $u+v+1$ is the critical value of $b_3$ such that there exists a sequence $A$ of positive integers for which $P(A)=\mathbb{N}\backslash \{u<v<b_3<\cdots\}$. In this paper, we obtain the critical value of $b_k$.