Relative density under fixed-parameter extrapolation

Starting from $0$ and $1$, repeatedly apply the operation $(a,b)\mapsto(1-\lambda)a+\lambda b$, with a fixed complex parameter $\lambda$. We prove that the generated set has bounded holes in the real line for every real $\lambda\notin[0,1]$, and in the complex plane for every nonreal $\lambda$. This answers the relative-density question of Fenner, Green and Homer. Their characterization by integer polynomials allows a single affine image of a relatively dense set of finite digit expansions to be placed inside the generated set.