Dense subsets of asymptotic bases

Let $h⩾2$ be an integer. A set $A$ of nonnegative integers is defined as an asymptotic basis of order $h$ if all sufficiently large integers can be expressed as a sum of $h$ elements from $A$. Write $d_L\left(A\right)$ as the lower asymptotic density of $A$ and $d_U\left(A\right)$ as the upper asymptotic density of $A$. In 1989, Nathanson and Sárközy proved that, if $A$ is an asymptotic basis of order $h$ and $B$ is a subset of $A$ with $d_L\left(B\right)>1/h$, then there exists a finite subset $F$ of $A$ such that $F\cup B$ is an asymptotic basis of order $h$. Recently, Xu and Chen [C. R. Math. Acad. Sci. Paris 362 (2024), 45-49.] further proved that, if $A$ is a set of nonnegative integers with $d_L\left(A\right)>0$, then there exists a subset $B$ of $A$ with $d_L\left(B\right)>0$ such that $F\cup B$ is not an asymptotic basis of order $h$ for any finite set $F$. In this paper, we improve the above result, that is: if $A$ is a set of nonnegative integers with $d_L\left(A\right)>0$, then there exists a subset $B$ of $A$ such that $d_L\left(B\right)⩾\frac{d_L\left(A\right)}{h-1+d_L\left(A\right)}\cdot d_L\left(A\right),d_U\left(B\right)=d_U\left(A\right),$and $F\cup B$ is not an asymptotic basis of order $h$ for any finite set $F$. We also show that in some sense the lower bound $\frac{d_L\left(A\right)}{h-1+d_L\left(A\right)}\cdot d_L\left(A\right)$ is the best possible.