On the restricted order of asymptotic bases

Abstract

Let $N$ be the set of all positive integers. For a set A of positive integers, let $A\sim N$ denote that A contains all but finitely many positive integers. For an integer $h⩾2$, define $hA=\{a_1+\cdots +a_h:a_1,\cdots ,a_h\in A\}$ and $h\times A=\{a_1+\cdots +a_h:a_1,\cdots ,a_h\in A,a_i\ne a_j$ for $i\ne j\}$. In 2023, Chen and Yu [Discrete Math. 346 (2023), Paper No. 113388.] proved that, there exists a set B of positive integers such that: ${\lim }_{x\to \infty }B\left(x\right)/x=1/2$, $B\bigcup \left(2B\right)\sim N$, $B\bigcup (2\times B)\nsim N$, and $B\bigcup (2\times B)\bigcup (3\times B)\sim N$. In this paper, we construct a somewhat dense set B satisfying the above properties. That is, there exists a set B of positive integers such that: ${\lim \inf }_{x\to \infty }B\left(x\right)/x=1/2$, ${\lim \sup }_{x\to \infty }B\left(x\right)/x=1$, $B\bigcup \left(2B\right)\sim N$, $B\bigcup (2\times B)\nsim N$, and $B\bigcup (2\times B)\bigcup (3\times B)\sim N$.