Minimal asymptotic bases related to G-adic sequences

Abstract: Let $A$ be a set of nonnegative integers and $h\ge 2$. The set $A$ is defined as an asymptotic basis of order $h$ if all sufficiently large integers $n$ can be expressed as the sum of $h$ elements taken from $A$. Such $A$ is further defined as \emph{minimal} if no proper subset of $A$ is an asymptotic basis of order $h$. In 1974, Nathanson explicitly constructed a minimal asymptotic basis of order $2$ by using binary representations. In 2022, Nathanson constructed a new class of minimal asymptotic bases of order $h$ based on the $\mathcal{G}$-adic sequence, where a $\mathcal{G}$-adic sequence $\mathcal{G}=\{g_i\}_{i=0}^{\infty}$ is a strictly increasing sequence of positive integers such that $g_0=1$ and $g_{i-1}$ divides $g_i$ for all $i\ge 1$. Recently, we improve the above result.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)