Sumsets of Ahlfors--David regular sets

We prove that there is an absolute constant $c>0$ such that for every finite $(α,M)$-Ahlfors--David regular set $A\subseteq[N]$ with $0\leqα<1$ and $M\geq2$, we have the sumset estimate \[|A+A|\geq |A|^{1+c(1-α)/\log M}.\] We also prove a corresponding statement for Ahlfors--David regular sets in $[