arXiv:2407.11961v3

On Fourier asymptotics and effective equidistribution

Shreyasi Datta, Subhajit Jana

math.DSmath.NT11J8328A80

Paper metadata

Original submission: July 16, 2024
Latest revision: July 12, 2026 · version 3
Claimed MSC: 11J83, 28A80
arXiv categories: math.DS, math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 30 pages, To appear in Crelle's Journal

Abstract

We prove effective equidistribution of expanding horocycles in $\mathrm{SL}_2(\mathbb{Z})\backslash\mathrm{SL}_2(\mathbb{R})$ with respect to various classes of Borel probability measures on $\mathbb{R}$ having certain Fourier asymptotics. Our proof involves new techniques combining tools from automorphic forms and harmonic analysis. In particular, for any Borel probability measure $μ$ , satisfying $\sum_{\mathbb{Z}\ni|m|\leq X}|\widehatμ(m)| = O\left(X^{1/2-θ}\right)$ with $θ>7/64,$ our result holds. This class of measures contains convolutions of $s$ -Ahlfors regular measures for $s>39/64$ , as well as a subclass of self-similar measures. Moreover, our result is sharp upon the Ramanujan--Petersson Conjecture (upon which the above $θ$ can be chosen arbitrarily small): there are measures $μ$ with $\widehatμ(ξ) = O\left(|ξ|^{-1/2+ε}\right)$ for which equidistribution fails.

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Audit summary

Audited against arXiv v3

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 15, 2026

01Statements4 reported findingsContains unsupported statements

The averaged Fourier criterion for effective horocycle equidistribution and the Khintchine-type application are supported. The full decay ranges asserted in Theorem 2 and Corollary 4 are not able to be verified because the remainder estimate used in their proof yields the smaller exponent $\delta_0-1/2$ , not $\delta_0$ .

Theorem 1 and Corollaries 1–3Correct

Effective equidistribution from averaged Fourier decay

Pages 2–3 and 21–27 · Theorem 1, Corollaries 1–3, and Section 5 · arXiv:2407.11961v3

The spectral expansion reduces the horocycle average to sums of automorphic Fourier coefficients weighted by $\widehat\mu(m)$ . Proposition 5.1 and Lemmas 5.1–5.3 control the cuspidal and Eisenstein contributions using the assumed average bound $\sum_{|m|\leq X}|\widehat\mu(m)|=O(X^{1/2-\theta})$ . The exceptional-eigenvalue and Ramanujan exponents are combined with the correct signs, giving the threshold $\theta>7/64$ . The Ahlfors-regular convolution and self-similar specializations supply exactly this hypothesis in the stated ranges.

Full paper, version 3 ↗

Theorem 2Not able to verify

The claimed full pointwise-asymptotic rate is not established

Page 4 and pages 27–29 · Theorem 2 and its proof · arXiv:2407.11961v3

The theorem assumes a remainder $O(|\xi|^{-\delta_0})$ with $\delta_0>1/2$ and permits every $0<\eta<\min\left\{\frac12,\delta_0,\frac{\delta_1}{2},\ldots,\frac{\delta_K}{2}\right\}.$ In the automorphic Fourier sum, that remainder is multiplied by the factor $|m|^{-1/2}$ , so it has denominator $|m|^{1+(\delta_0-1/2)}$ . Lemma 6.1 therefore supplies only $\eta<\min\{1/2,\delta_0-1/2\}$ for this term. No other argument recovers the larger printed range. The expansion may hold with a stronger rate, but the supplied proof does not verify it.

Full paper, version 3 ↗

Corollary 4Not able to verify

The claimed rate does not follow from decay just beyond one half

Page 4 · Corollary 4; pages 28–29 · proof of Theorem 2 · arXiv:2407.11961v3

If $\widehat\mu(\xi)=O(|\xi|^{-(1/2+\delta)})$ , the remainder calculation in Theorem 2 gives effective equidistribution for $\eta<\min\{1/2,\delta\}$ . Corollary 4 instead asserts the conclusion for every $\eta<1/2$ , regardless of the fixed positive $\delta$ . For example, when $0<\delta<1/2$ , Lemma 6.1 does not control the remainder at rates $\delta<\eta<1/2$ . No independent argument for those rates is supplied.

Theorem 3Correct

Khintchine-type divergence conclusion

Pages 5 and 24–27 · Theorem 3, Proposition 5.2, and Lemma 5.4 · arXiv:2407.11961v3

Proposition 5.2 gives the uniform effective equidistribution input required by the cited shrinking-target theorem, and Lemma 5.4 verifies the regularity and nonconcentration hypotheses for the measure class used here. Applying that theorem to the divergent approximation function yields the stated full-measure conclusion. The truncation and monotonicity reductions preserve every admissible denominator and do not introduce an unhandled convergence case.

02Proofs5 reported findingsContains incorrect or incomplete proofs

The proof of the averaged Fourier criterion and the Diophantine application is correct. Theorem 2 and Corollary 4 use an unsupported decay rate, and the zero-free case of Corollary 5 is omitted; the latter has a verified nonstationary-phase repair.

Proof of Theorem 1Correct and complete

Spectral and Fourier estimates

Pages 21–27 · Section 5 · arXiv:2407.11961v3

The constant term produces the Haar average. The nonconstant cuspidal and Eisenstein modes are split at the appropriate frequency scale, after which the average Fourier hypothesis, Rankin--Selberg bounds, and the spectral-gap exponent control all pieces. The smoothing loss is balanced against those estimates, and the inequality $\theta>7/64$ makes every residual exponent positive.

Proof of Theorem 2Incorrect as written

The remainder term is passed to Lemma 6.1 with an exponent larger by one half

Pages 20 and 27–29 · Lemma 6.1 and proof of Theorem 2 · arXiv:2407.11961v3

The remainder $O(|m|^{-\delta_0})$ appears in a Fourier series already carrying $|m|^{-1/2}$ , hence contributes a sum weighted by $|m|^{-1-(\delta_0-1/2)}.$ Lemma 6.1, whose hypothesis is a denominator $|m|^{1+\delta}$ , can therefore be used with $\delta=\delta_0-1/2$ , not with $\delta=\delta_0$ . Repair classification: Verified repair. Replace the permitted range in Theorem 2 by $0<\eta<\min\left\{\frac12,\delta_0-\frac12,\frac{\delta_1}{2},\ldots,\frac{\delta_K}{2}\right\}.$ Every subsequent estimate then follows exactly as written.

Full paper, version 3 ↗

Proof of Corollary 4Incorrect as written

The corollary discards the decay margin that controls the rate

Page 4 and pages 28–29 · Corollary 4 and the remainder argument · arXiv:2407.11961v3

Substituting $\delta_0=1/2+\delta$ into the corrected remainder estimate gives $\eta<\min\{1/2,\delta\}$ . The printed proof instead retains only $\eta<1/2$ . Repair classification: Verified repair. Add the condition $\eta<\delta$ to Corollary 4, or strengthen its Fourier-decay hypothesis to an exponent exceeding $1$ if all $\eta<1/2$ are desired. No other part of the proof changes.

Proof of Corollary 5Incomplete as written

The zero-free case is omitted

Page 5 and pages 27–28 · definition of $k_{f,w}$ , Proposition 6.1, and proof of Corollary 5 · arXiv:2407.11961v3

A nonconstant analytic function may have no critical point on $\operatorname{supp}(w)$ ; for example, $f(x)=x$ is allowed. Then the printed maximum defining $k_{f,w}$ is over an empty set, and Proposition 6.1 and the one-line proof do not cover this case. Repair classification: Verified repair. Define $k_{f,w}=0$ when the critical set is empty. Since $|f'|$ is then bounded below on the compact support, repeated integration by parts gives $\widehat\mu_{w,f}(\xi)=O_N((1+|\xi|)^{-N})$ for every $N$ ; applying Theorem 2 with an arbitrarily large remainder exponent yields Corollary 5 for every $0<\eta<1/2$ , exactly the range given by $k_{f,w}=0$ .

Proof of Theorem 3Correct and complete

Effective shrinking-target application

Pages 24–27 · Proposition 5.2, Lemma 5.4, and proof of Theorem 3 · arXiv:2407.11961v3

The effective equidistribution bound is uniform over the smooth approximations to the shrinking targets, and the derivative costs are included in the selected smoothing scale. Lemma 5.4 supplies the required local dimension estimate. The cited divergence theorem then applies to the monotone approximation function, and the standard dyadic reduction recovers the full sequence.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2407.11961v3
Authors listed: Shreyasi Datta, Subhajit Jana
Audit date: August 15, 2026

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On Fourier asymptotics and effective equidistribution

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Effective equidistribution from averaged Fourier decay

The claimed full pointwise-asymptotic rate is not established

The claimed rate does not follow from decay just beyond one half

Khintchine-type divergence conclusion

Spectral and Fourier estimates

The remainder term is passed to Lemma 6.1 with an exponent larger by one half

The corollary discards the decay margin that controls the rate

The zero-free case is omitted

Effective shrinking-target application

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