John and uniform domains in generalized Siegel boundaries

Given the pair of vector fields $X={\mathrm{\partial }}_x+\mathrm{\mid }z{\mathrm{\mid }}^{2m}y{\mathrm{\partial }}_t$ and $Y={\mathrm{\partial }}_y-\mathrm{\mid }z{\mathrm{\mid }}^{2m}x{\mathrm{\partial }}_t,$ where $(x,y,t)= (z,t)\in$\mathbb{R}$^3=$\mathbb{C}$\times$\mathbb{R}$$, we give a condition on a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ which ensures that $\mathrm{\Omega }$ is an $(ϵ,\delta )$-domain for the Carnot-Carath'eodory metric. We also analyze the Ahlfors regularity of the natural surface measure induced at the boundary by the vector fields.