On the minimal length of addition chains

We denote by $\mathrm{\ell }(n)$ the minimal length of an addition chain leading to $n$ and we define the counting function $$F(m,r):=\mathrm{\#}\{n\in [2^m,2^{m+1}):\mathrm{\ell }(n)\le m+r\},$$ where $m$ is a positive integer and $r\ge 0$ is a real number. We show that for $0<c<\log 2$ and for any $\epsilon >0$, we have as $m\to \mathrm{\infty }$, $$F(m,\frac{cm}{\log m})<\exp (cm+\frac{\epsilon m\log \log m}{\log m})$$ and $$F(m,\frac{cm}{\log m})>\exp (cm-\frac{(1+\epsilon )cm\log \log m}{\log m})\mathrm{.}$$ This extends a result of Erdős which says that for almost all $n$, as $n\to \mathrm{\infty }$, $$\mathrm{\ell }(n)=\frac{\log n}{\log 2}+(1+o(1))\frac{\log n}{\log \log n}\mathrm{.}$$