Is an asymptotic equivalent known or conjectured for the number $N(d)$ of $\alpha \in \bar{\mathbb{Q}}$ with $h(\alpha) < 1$ and $[\mathbb{Q}(\alpha):\mathbb{Q}] \leq d$? 

The rather crude proof of Northcott's theorem bounds $\log{N(d)}$ by $O(d^2)$. On the other hand, upon extracting $\lfloor d/m \rfloor$-th roots from numbers of degree $m$ and logarithmic height $< d/m$, where $m$ is fixed but arbitrarily large, and using a generalization of Schanuel's theorem due to Masser and Vaaler ("Counting algebraic numbers with large height," *Trans. Amer. Math. Soc.* 2007), it is easily seen that $\log{N(d)} > Ad$ for $d \gg_A 0$ and any given $A < \infty$. 

The mentioned paper of Masser and Vaaler determines the dominant term for the opposite count involving points of large height and bounded degree. The problem considered here, involving bounded height and large degree, is likely to be more delicate. The basic lower bound $\log{N(d)} \gg d$ of the previous paragraph is likely to be improved by taking an optimal $m = m(d)$ and examining the error term (rate of convergence) in the Masser-Vaaler estimate. Perhaps such an improvement could point to the right asymptotics for $\log{N(d)}$?

Has this asymptotic been studied?