Limit Points of Nathanson’s Lambda Sequences

Abstract

We consider the set $A_{n}=\displaystyle \cup _{j=0}^{\infty }\{\varepsilon _{j}(n)\cdot n^j:\varepsilon _{j}(n)\in \{0,\pm 1,\pm 2,\ldots ,\pm \lfloor {{n}/{2}}\rfloor \}\}$. Let $\mathscr {S}_{\mathscr {A}}= \bigcup _{a \in \mathscr {A} } A_{a}$ where $\mathscr {A}\subseteq \mathbb {N}$. We denote by $\lambda _{\mathscr {A}}(h)$ the smallest positive integer that can be represented as a sum of h, and no less than h, elements in $\mathscr {S}_{\mathscr {A}}$. Nathanson studied the properties of the $\lambda _\mathscr {A}(h)$-sequence and posed the problem of finding the values of $\lambda _\mathscr {A}(h)$. When $\mathscr {A}=\{2,i\}$, we represent $\lambda _{\mathscr {A}}(h)$ by $\lambda _{2,i}(h)$. Only the values $\lambda _{2,3}(1)=1$, $\lambda _{2,3}(2)=5$, $\lambda _{2,3}(3)=21$ and $\lambda _{2,3}(4)=150$ are known. In this paper, we extend this result. For odd $i>1$ and $h\in \{1,2,3\}$, we find an extensive set of values for $\lambda _{2,i}(h)$. Furthermore, for fixed $h\in \{1,2,3\}$, we find certain 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 limit points of Nathanson’s lambda sequences.