arXiv:2312.06860v2
Construction of Brooks-Lindenstrauss kernels on affine buildings of arbitrary reduced type, with applications
math.CAmath.DSmath.GRmath.NTmath.RT81Q5011F8537A4437D4022E4535P20
Abstract
This article deals with harmonic analysis on affine buildings. Its main goal is to construct suitable kernels associated to a discrete multitemporal wave equations on the latter spaces, the long-standing motivation being to contribute to progress in arithmetic quantum unique ergodicity (AQUE) on certain Riemannian manifolds.
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01Statements2 reported findingsCorrect
Both advertised results are supported. Theorem A/3.1 supplies finitely supported radial kernels whose transforms concentrate at a prescribed spectral parameter and whose spatial coefficients decay exponentially. Theorem B/4.1 applies these kernels to an A-invariant weak limit of joint eigenfunctions and proves that every nonzero ergodic component has positive entropy for a regular element.
Theorem A / Theorem 3.1Correct
Localized Brooks–Lindenstrauss kernels on affine buildings
Pages 2 and 11–17 · Theorem A / Theorem 3.1 · arXiv:2312.06860v2
For every spectral parameter in the stated strip, Theorem A constructs a real radial kernel with the announced finite propagation, lower bound at the target parameter, uniform transform bound, and exponential spatial decay. The integers M and N are allowed to depend only on the real part of the target parameter as stated, while the support radius scales linearly with N. The construction is carried out in the spherical Hecke algebra appropriate to the fixed building, so its operator interpretation matches the later entropy application.
Theorem B / Theorem 4.1Correct
Positive entropy of nonzero ergodic components
Pages 3 and 18–24 · Theorem B / Theorem 4.1 · arXiv:2312.06860v2
The theorem assumes the R-split semisimple setting, a sequence of normalized joint eigenfunctions with parameters in the stated compact regular set, and an A-invariant weak limit. For any regular a in the chosen Weyl chamber, the proof gives positive Kolmogorov–Sinai entropy on each ergodic component to which the limiting measure assigns positive mass. The kernel constants are uniform over the compact parameter set, so the conclusion applies componentwise rather than only to the averaged measure.
02Proofs3 reported findingsCorrect
The harmonic-analysis construction and the Shem-Tov argument. The proof selects a simultaneous Dirichlet approximation so that a Fejér-type polynomial peaks at the target eigenvalue. Applying this polynomial to the spherical averaging operator gives finite propagation from its degree. The inverse Fourier–Gelfand formula and the building's spherical-function bounds yield exponential decay in distance, while the target phase choice gives the required positive lower bound. The estimates separate the tempered boundary from the complementary strip and choose M before N, exactly as the quantified statement requires.
Kernel construction and entropy transferCorrect and complete
The harmonic-analysis construction and the Shem-Tov argument
Pages 11–24 · Sections 3–4 · arXiv:2312.06860v2
The proof selects a simultaneous Dirichlet approximation so that a Fejér-type polynomial peaks at the target eigenvalue. Applying this polynomial to the spherical averaging operator gives finite propagation from its degree. The inverse Fourier–Gelfand formula and the building's spherical-function bounds yield exponential decay in distance, while the target phase choice gives the required positive lower bound. The estimates separate the tempered boundary from the complementary strip and choose M before N, exactly as the quantified statement requires.
Section 4Correct and complete
Entropy transfer from kernel bounds
Pages 18–24 · proof of Theorem 4.1 · arXiv:2312.06860v2
The kernel operator is applied to small Bowen sets and its spectral lower bound prevents an eigenfunction from concentrating on too few orbit names. Finite propagation bounds the number of cells that can interact, while exponential coefficient decay makes the contribution of distant cells summable. The covering estimate therefore gives exponential growth of distinguishable names on every positive-mass ergodic component. The Brin–Katok/entropy criterion is invoked for the regular element fixed in the theorem, and the zero component is explicitly excluded, giving exactly the positive-entropy conclusion.
Cross-referenceTypo · no status impact
A theorem is called a corollary
Page 12 · line preceding Equation (15) · arXiv:2312.06860v2
The text says ‘according to Corollary 2.3’, while the relevant earlier result is Theorem 2.3. The number and mathematical content uniquely identify the intended reference, so this literal label error does not affect the argument.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.