Limit points of Nathanson's Lambda sequences

Abstract

We consider the set $A_{n}=\displaystyle\cup_{j=0}^{\infty}\{\varepsilon_{j}(n)\cdot n^j\colon\varepsilon_{j}(n)\in\{0,\pm1,\pm2,...,\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 $\textit{limit points of Nathanson's lambda sequences}$.