Logarithmic growth conjecture for the set T(α)

Let $T(\alpha)$ be the set defined in the paper, and write $T_{\leq x}(\alpha)$ for its elements up to $x$. Logarithmic growth conjecture. For suitable $\alpha\in(0,1)$, there exists $\lambda=\lambda(\alpha)>0$ such that

$$\# T_{\leq x}(\alpha)=\lambda\log x+o(\log x)\quad\text{as }x\to\infty.$$

This predicts a positive logarithmic asymptotic for the counting function of $T(\alpha)$, strengthening the preceding result that $T(\alpha)$ is non-empty for every $\alpha\in(0,1/9)$. The meaning of “suitable” and the precise definition of $T(\alpha)$ should be checked in the paper.