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 $\ell^1$-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 $\kappa/(\kappa+s)$, where $\kappa$ 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.