On Minimal Asymptotic Bases
Cui-Fang Sun, Cheng Zhi · Mathematical Notes · 2022
Let $$\mathbb N$$ denote the set of all nonnegative integers, and let $$A\subseteq\mathbb N$$ . Let $$h,n\in\mathbb N$$ , $$h\ge 2$$ and $$r_h(A,n)=\#\{(a_1,\dots,a_h)\in A^h:a_1+\dotsb+a_h=n\}$$ . The set $$A$$ is called an asymptotic basis of order $$h$$ if $$r_h(A,n)\ge 1$$ for all sufficiently large integer $$n$$ . An asymptotic basis $$A$$ of order $$h$$ is minimal if no proper subset of $$A$$ is an asymptotic basis of order $$h$$ . Recently, Sun used 2-adic representations of integers to construct a new class of minimal asymptotic bases of order $$h$$ . In this paper, we generalize the 2-adic result to the $$g$$ -adic case.