Box-counting dimensions of popcorn subsets (Q2698059)

Let \(S\) be a subset of \(\mathbb N\). Let \[ G_S=\left\{\left(\frac{p}{q},\frac{1}{q}\right):\gcd(p,q)=1,\,1\le p<q,\,q\in S\right\} \] and \[ F_S=\left\{\left(\frac{p}{q},\frac{1}{q}\right):1\le p<q,\,q\in S\right\}. \] \textit{H. Chen} et al. [Proc. Am. Math. Soc. 150, No. 11, 4729--4742 (2022; Zbl 07594306)] proved that the popcorn sets \(G=G_{\mathbb N}\) and \(F=F_{\mathbb N}\) are of Assouad dimension \(2\) and box-counting dimension \(4/3\) by using estimates from number theory and probability. In this paper, by introducing logarithm density and \(\beta\)-condition for \(S\), upper and lower bounds of boxcounting dimensions of some popcorn subsets \(S\) are given.