Sumsets of Ahlfors--David regular sets

We prove that there is an absolute constant $c>0$ such that for every finite $(\alpha ,M)$-Ahlfors–David regular set $A\subseteq [N]$ with $0\le \alpha <1$ and $M\ge 2$, we have the sumset estimate $$\mathrm{\mid }A+A\mathrm{\mid }\ge \mathrm{\mid }A{\mathrm{\mid }}^{1+c(1-\alpha )\mathrm{/}\log M}\mathrm{.}$$ We also prove a corresponding statement for Ahlfors–David regular sets in $[0,1]$. The proof combines a multiscale entropy decomposition with inverse results from additive combinatorics, showing that Ahlfors–David regularity forces a definite entropy gain at each scale. The dependence $1\mathrm{/}\log M$ in the exponent is of optimal order.