Limit points of Nathanson's Lambda sequences

Abstract

We consider the set $A_n={\cup }_{j=0}^{\infty }\{{\epsilon }_j(n)\cdot n^j:{\epsilon }_j(n)\in \{0,\pm 1,\pm 2,...,\pm \lfloor n/2\rfloor \}\}$. Let ${\mathcal{S}}_{\mathcal{A}}={\bigcup }_{a\in \mathcal{A}}{A}_a$ where $\mathcal{A}\subseteq \mathbb{N}$. We denote by ${\lambda }_{\mathcal{A}}(h)$ the smallest positive integer that can be represented as a sum of $h$, and no less than $h$, elements in ${\mathcal{S}}_{\mathcal{A}}$. Nathanson studied the properties of the ${\lambda }_{\mathcal{A}}(h)$-sequence and posed the problem of finding the values of ${\lambda }_{\mathcal{A}}(h)$. When $\mathcal{A}=\{2,i\}$, we represent ${\lambda }_{\mathcal{A}}(h)$ by ${\lambda }_{2,i}(h)$. Only the values ${\lambda }_{2,3}(1)=1$, ${\lambda }_{2,3}(3)=5$, ${\lambda }_{2,3}(3)=21$ and ${\lambda }_{2,3}(4)=150$ are known. In this paper, we extend this result. For any odd $i>1$ and $h\in \{1,2,3\}$, we find the values of ${\lambda }_{2,i}(h)$. Furthermore, for fixed $h\in \{1,2,3\}$, we find the values of ${\lambda }_{2,i}(h)$ that occur infinitely many times as $i$ runs over the odd integers bigger than 1. We call these numbers the $\text{limit points of Nathanson’s lambda sequences}$.

10 pages, 2 tables