Prescribed Beurling densities in spectral eigen-subspaces
Abstract
Let (q,D,L) be a one-dimensional Hadamard triple with 2≤ m:=#D<q, and let μq,D be the associated self-similar probability measure. Write 𝒯={t∈ℤ:(q,D,tL) is a Hadamard triple}. We prove that, for every 0<s<log m/log q and every ζ∈[0,+∞], there are continuum many spectra Λ of μq,D such that tΛ is a spectrum for every t∈𝒯, dimBeΛ=s, and Ds+(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.
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BibTeX
@misc{lu2026prescribed,
title = {{Prescribed Beurling densities in spectral eigen-subspaces}},
author = {Zheng-Yi Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26146},
eprint = {2610.26146},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26146},
note = {CC0},
}
RIS
TY - UNPB
TI - Prescribed Beurling densities in spectral eigen-subspaces
AU - Zheng-Yi Lu
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26146
UR - https://arxiv.org/abs/2610.26146
AB - Let (q,D,L) be a one-dimensional Hadamard triple with 2≤ m:=#D<q, and let μ_q,D be the associated self-similar probability measure. Write 𝒯={t∈ℤ:(q,D,tL) is a Hadamard triple}. We prove that, for every 0<s<log m/log q and every ζ∈[0,+∞], there are continuum many spectra Λ of μ_q,D such that tΛ is a spectrum for every t∈𝒯, dim_BeΛ=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.
ER -