Wiener's lemma for quasicrystals: hull-uniform decay of covariant kernels on Delone sets
Abstract
This paper studies continuous covariant kernel families over hulls of Delone sets of finite local complexity using graded Wiener-algebra methods. The norm given by the least common weighted summable off-diagonal profile is identified isometrically with a graded ℓ1-norm. The resulting Banach *-algebras are symmetric, spectrally invariant in the reduced groupoid C*-algebra, and stable under holomorphic functional calculus. Exact physical weights give preservation of polynomial and stretched-exponential classes, including a fixed stretched-exponential rate. A configuration with dense orbit has a faithful regular representation. Invertibility on that configuration therefore gives an inverse family with one profile valid throughout the hull; for repetitive sets every configuration has this property. For polynomial weights of order s>0, return-vector counting yields a differential estimate with parameter κ/(κ+s), where κ is a growth exponent of the return vectors and may be taken equal to d for Meyer sets. Combined with established norm-controlled inversion criteria, this gives explicit bounds in the common-profile norm. Explicit bounds are also obtained for stretched-exponential physical weights and for gapped Fermi projections.
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BibTeX
@misc{jaure2026wieners,
title = {{Wiener's lemma for quasicrystals: hull-uniform decay of covariant kernels on Delone sets}},
author = {Diego Jaure},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12770},
eprint = {2610.12770},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12770},
}
RIS
TY - UNPB
TI - Wiener's lemma for quasicrystals: hull-uniform decay of covariant kernels on Delone sets
AU - Diego Jaure
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12770
UR - https://arxiv.org/abs/2610.12770
AB - This paper studies continuous covariant kernel families over hulls of Delone sets of finite local complexity using graded Wiener-algebra methods. The norm given by the least common weighted summable off-diagonal profile is identified isometrically with a graded ℓ^1-norm. The resulting Banach *-algebras are symmetric, spectrally invariant in the reduced groupoid C*-algebra, and stable under holomorphic functional calculus. Exact physical weights give preservation of polynomial and stretched-exponential classes, including a fixed stretched-exponential rate. A configuration with dense orbit has a faithful regular representation. Invertibility on that configuration therefore gives an inverse family with one profile valid throughout the hull; for repetitive sets every configuration has this property. For polynomial weights of order s>0, return-vector counting yields a differential estimate with parameter κ/(κ+s), where κ is a growth exponent of the return vectors and may be taken equal to d for Meyer sets. Combined with established norm-controlled inversion criteria, this gives explicit bounds in the common-profile norm. Explicit bounds are also obtained for stretched-exponential physical weights and for gapped Fermi projections.
ER -