A sufficient condition for full density of $V(1,1)$

Let $V(1,1)$ be the visibility set considered in the paper. For $1\leqslant a,b\leqslant N$, let $b_1<b_2<\dots<b_{k_a}$ be the values of $b$ such that $(a,lb_i)\notin V(1,1)$ for every $l\in\mathbb{N}$ with $1\leqslant l\leqslant N/b_i$, and define

$$c(a):=\frac{n(a,N)}{N},$$

where $n(a,N)$ is the number of $b$ with $(a,b)\notin V(1,1)$. Set

$$c_1(a):=\sum_{i=1}^{k_a}\frac{1}{b_i}.$$

Full-density criterion for $V(1,1)$. If

$$\sum_{a=1}^N c_1(a)=o(N),$$

then $\operatorname{dens}(V(1,1))=1$.