arXiv:2601.21954v2

Asymptotic Expansion for Expanding Spherical Averages in Real Rank One

Zhiyuan Deng, Yutian Sun

math.RT22E4622E4022F3037A17

Paper metadata

Original submission: January 29, 2026
Latest revision: June 4, 2026 · version 2
Claimed MSC: 22E46, 22E40, 22F30, 37A17
arXiv categories: math.RT
Paper license: arXiv non-exclusive distribution license ↗

Abstract

This paper establishes an asymptotic expansion for expanding sector average on compact quotients of real rank-one Lie groups, with $SO(n,1)$ as a model case. Using tools from harmonic analysis and representation theory, the problem is reduced, via the action of the Casimir operator, to an ordinary differential equation(ODE) governing the evolution along the expanding $A$ -direction. The asymptotic expansion is then derived from the analysis of this ODE.

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsContains wrong statements

The main asymptotic-expansion theorem is false as stated because it omits the invariant constant component of the test function. The error is exposed by the constant function and also appears in the general vector statement used to prove the theorem.

Theorem 1.1Incorrect

The expansion omits the nondecaying constant term

Pages 2–3 · Theorem 1.1 · arXiv:2601.21954v2

The theorem is stated for every $f\in W^\beta(X)$ , while all displayed exponents in the claimed expansion have negative real part in the compact spectral-gap setting and the remainder decays. Set $f\equiv1$ and choose $\phi$ with $\int_K\phi\neq0$ . Then the spherical average is identically $I(t,1,\phi)=\int_K\phi(k)\,dk,$ which does not decay. The statement becomes valid after adding the constant term $(\int_X f)(\int_K\phi)$ and applying the expansion to $f-\int_Xf$ , or after imposing mean zero.

Theorem 6.5Incorrect

The abstract expansion drops two hypotheses needed by its input proposition

Pages 45–47 · Proposition 6.2 and Theorem 6.5 · arXiv:2601.21954v2

Proposition 6.2 assumes that $v$ is orthogonal to constants and belongs to $W^{\beta,2}$ , and its remainder is controlled by $\|v\|_{W^{\beta,2}}$ . Theorem 6.5 is stated for arbitrary $v\in W^{k,2}$ and neither removes the invariant vector nor assumes the regularity appearing in the estimate. Taking the invariant vector gives the same direct contradiction as $f\equiv1$ .

Proposition 4.3 and Theorem 5.23Correct

The joint-eigenvector expansion is supported

Pages 20–21 and 42–44 · arXiv:2601.21954v2

The Iwasawa-coordinate Casimir identity gives the displayed second-order inhomogeneous ODE for a joint $(\Omega_G,\Omega_M)$ eigenvector after integration by parts on $K$ . Iterating its two fundamental solutions produces the finite exponential-polynomial expansion, and Lemmas 5.20–5.22 give the uniform remainder and Sobolev regularity of its coefficients. This part verifies the expansion on each spectral component; the defect arises only when Section 6 extends it beyond its mean-zero and regularity hypotheses.

02Proofs2 reported findingsContains incorrect or incomplete proofs

The representation-theoretic expansion is derived only on the orthogonal complement of constants, but the proof applies it to arbitrary functions and also weakens the Sobolev hypothesis without justification. The printed proof therefore does not establish either Theorem 6.5 or Theorem 1.1.

Passage from Proposition 6.2 to Theorem 6.5Incorrect as written

Mean-zero and Sobolev assumptions are lost

Pages 45–47 · Section 6 · arXiv:2601.21954v2

The spectral expansion of the nontrivial representation is valid after projecting away the invariant line. The proof never performs that projection when it states Theorem 6.5, and it replaces the $W^{\beta,2}$ hypothesis by the weaker $W^{k,2}$ hypothesis even though the displayed remainder still contains $\|v\|_{W^{\beta,2}}$ . A repair must split $v=(\int v)\mathbf1+v_0$ , retain the first summand exactly, and require the same Sobolev order for $v_0$ as in Proposition 6.2.

Propositions 6.2 and 6.4Correct and complete

Spectral summation is valid under the stated input hypotheses

Pages 45–46 · Section 6.1 · arXiv:2601.21954v2

For a mean-zero vector in $W^{\beta,2}$ , the Weyl bound and Cauchy–Schwarz make both the main-term and remainder sums Cauchy in $W^{k,2}$ . The exponent in the residual spectral sum is negative precisely because $\beta$ is chosen above the displayed threshold. Thus the componentwise estimates do sum correctly on the domain of Propositions 6.2 and 6.4; the proof fails only when Theorem 6.5 drops that domain.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2601.21954v2
Authors listed: Zhiyuan Deng, Yutian Sun
Audit date: August 18, 2026

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Asymptotic Expansion for Expanding Spherical Averages in Real Rank One

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The expansion omits the nondecaying constant term

The abstract expansion drops two hypotheses needed by its input proposition

The joint-eigenvector expansion is supported

Mean-zero and Sobolev assumptions are lost

Spectral summation is valid under the stated input hypotheses

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