Prescribed Beurling densities in spectral eigen-subspaces

Source abstract

Let $(q,D,L)$ be a one-dimensional Hadamard triple with $2\leq m:=\#D<q$, and let $μ_{q,D}$ be the associated self-similar probability measure. Write $\mathcal T=\{t\in\mathbb Z:(q,D,tL)\text{ is a Hadamard triple}\}$. We prove that, for every $0<s<\log m/\log q$ and every $ζ\in[0,+\infty]$, there are continuum many spectra $Λ$ of $μ_{q,D}$ such that $tΛ$ is a spectrum for every $t\in\mathcal T$, $\dim_{\mathrm{Be}}Λ=s$, and $D_s^+(Λ)=ζ$. The construction combines a nested family of finite digit sets with lacunary spectral extensions. A two-sided interval-counting estimate controls the density of the prescribed subset, while the added frequencies ensure completeness for every multiplier without changing that density. Binary choices in the extension yield continuum many distinct spectra. This refines Beurling dimension interpolation in spectral eigen-subspaces by prescribing the upper density at every strictly intermediate dimension.