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Quantitative Fourier decay for Patterson-Sullivan measures of dimension larger than $1/2$

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Pith's one-line read For every convex co-compact Schottky group with limit-set dimension δ>1/2, the Patterson–Sullivan measure decays in Fourier space at the explicit polynomial rate |ξ|^{-δ(2δ−1)/((2δ+1)(3−δ))}, and the proof needs no additive-combinatorial ma

desk verdict Genuinely new explicit Fourier decay exponent for Patterson-Sullivan measures, but the central L2 lemma has an algebra error in the dyadic summation that needs fixing before the proof is complete. read the letter →

arxiv 2607.18010 v1 pith:IQZ3K4WD submitted 2026-07-20 math.DS math.CAmath.MG

Félix Lequen , Tuomas Sahlsten This is my paper · ORCID
classification math.DSmath.CAmath.MG MSC 37C4528A80
keywords FourierdecayPatterson–SullivanmeasureSchottkygroupconvexco-compactoscillatoryintegralstransposenon-concentrationAhlforsregularity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes an explicit polynomial decay rate for the Fourier transform of Patterson–Sullivan measures attached to convex co-compact Schottky groups, whenever the Hausdorff dimension of the limit set exceeds $\frac{1}{2}$. Previous proofs of Fourier decay in this setting relied on heavy tools such as additive combinatorics, renewal theory, transfer-operator techniques, or $L^2$-flattening; this paper replaces them with oscillatory integral estimates, hyperbolic geometry, and a duality argument based on the transpose Schottky group. If correct, the result gives a concrete, computable exponent and clarifies why the half-dimension threshold appears. The proof's key non-concentration estimate is a counting bound for how often ratios of matrix coefficients cluster, obtained from Ahlfors regularity of the Patterson–Sullivan measure of the transpose group.

What carries the argument

The transpose Schottky group $\Gamma^T$ and its Ahlfors $\delta$-regular Patterson–Sullivan measure $\nu$. The central object is the non-concentration estimate (Lemma 2.9): for any set of words with cylinder lengths comparable to $\tau$, the number of words whose ratio $c_a/d_a$ lies within $\sigma$ of a fixed point is at most $C \tau^{-\delta} \sigma^\delta$ for $\sigma \ge \tau$. This estimate controls near-degenerate pairs of cylinders in the $L^2$ norm of the oscillatory sum. The other main ingredients are the stopping-time partition $W_\tau$, bounded distortion (derivatives comparable to cylinder sizes), a Frostman approximation lemma for measures, and oscillatory integral estimates for phases with controlled second derivative and at most one critical point.

What would settle it

Two concrete tests: (1) For a non-symmetric word such as $a=12$ in a two-generator Schottky group, check the printed identity $\gamma^T_{a_n} \cdots \gamma^T_{a_1}(0) = J \gamma_{a_n \cdots a_1}(\infty)$ in Lemma 2.8 without the inverse overlines; it will fail, so one must verify numerically that the intended reversed-word cylinder comparison $|I_{\bar{a}}| \asymp |I_a|$ holds for all words—if the ratio is unbounded, the non-concentration estimate collapses. (2) For a Schottky group with $\delta \approx 0.6$, numerically estimate $|\hat{\mu}(\xi)|$ at large $\xi$; if it decays slower than $|\xi|^{-\frac{\delta(2\delta-1)}{(2\delta+1)(3-\delta)}} \approx |\xi|^{-0.023}$, Theorem 1 fails, while much faste

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Extended reading notes

Core claim

The paper proves that for any convex co-compact Schottky group $\Gamma \subset \mathrm{PSL}_2(\mathbb{R})$ with critical exponent $\delta > \frac{1}{2}$, the Fourier transform of its Patterson–Sullivan measure satisfies $|\hat{\mu}(\xi)| \leq C |\xi|^{-\frac{\delta(2\delta-1)}{(2\delta+1)(3-\delta)}}$ for all $|\xi| \ge 1$, with $C$ depending only on the group. The argument uses a stopping-time partition of the limit set, bounds the $L^2(\text{Lebesgue})$ norm of the resulting oscillatory sum in terms of the frequency $|\xi|$ and the scale $\tau$, and then optimizes $\tau = |\xi|^{-\frac{1}{2\delta+1}}$. The non-degeneracy of the phases is controlled by a counting estimate for the transpose group $\Gamma^T$, whose Patterson–Sullivan measure is Ahlfors $\delta$-regular. The exponent is slightly better than the earlier explicit exponent obtained

Load-bearing premise

The proof hinges on a counting estimate (Lemma 2.9) that assumes, for every reversed word, the cylinder size of the reversed word is comparable to the cylinder size of the original word; as printed, the derivation of this comparison in Lemma 2.8 omits the inverse letters and is false as written, so the whole argument leans on the intended corrected statement rather than on the printed identity.

Editorial extensions

If this is right

  • If Theorem 1 is correct, every convex co-compact Schottky group of dimension δ>1/2 has a Patterson–Sullivan measure with explicit polynomial Fourier decay, with the exponent determined solely by δ.
  • Via the fractal uncertainty principle, this yields an explicit improvement of the essential spectral gap of the Laplacian on the associated hyperbolic surface, though the paper notes the improvement is weak near δ=1/2.
  • The proof shows that additive-combinatorial, renewal, transfer-operator, and L2-flattening techniques are not needed for δ>1/2, opening a simpler route to quantitative Fourier decay in other conformal settings.
  • The method is stated to extend to Ahlfors-regular self-conformal measures satisfying a non-linearity condition on the dual IFS, and to geometrically finite groups with parabolic elements when δ≥1/2.
  • The obtained exponent improves slightly on the earlier explicit Gauss-map exponent because of a sharper approximation lemma.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the corrected reversed-word comparison in Lemma 2.8 holds, the same non-concentration counting argument should transfer to any conformal IFS whose dual IFS is Ahlfors δ-regular with δ>1/2, yielding explicit Fourier decay for a much larger class of self-conformal measures.
  • The threshold δ=1/2 appears exactly where the optimized scale τ = |ξ|^{-1/(2δ+1)} balances the L² term |ξ|^{-1/2}τ^{-1/2} with the τ^δ term; the proven exponent tends to 0 as δ→1/2+, so the theorem degrades continuously and a different mechanism would be needed at or below δ=1/2.
  • The dual-IFS interpretation (the dual action sends a family of functions to the transpose group) suggests that the non-concentration condition is equivalent to the non-linearity of the dual action; this may connect the result to existing separation-of-cylinders bounds for self-conformal sets.
  • A numerical check of the Fourier dimension of the Patterson–Sullivan measure for a simple two-generator Schottky group with δ near 0.6 could indicate whether the proven exponent is close to sharp or whether the approximation steps leave room for improvement.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper claims an elementary proof that Patterson–Sullivan measures of convex co-compact Schottky groups of dimension δ > 1/2 have polynomial Fourier decay with the explicit exponent δ(2δ−1)/((2δ+1)(3−δ)). The proof approximates the measure by sums over stopping-time cylinders, establishes an L^2(Lebesgue) estimate for the approximating function f_ξ using oscillatory integral bounds and a non-concentration lemma for the transpose Schottky group, and then returns to the measure via a Frostman-type averaging argument. The argument is intended to avoid sum–product estimates, renewal theory, Dolgopyat methods, and L^2 flattening used in earlier work.

Significance. If correct, this is a valuable contribution: it would give the first explicit polynomial Fourier decay exponent for Patterson–Sullivan measures of convex co-compact Schottky groups in the δ > 1/2 regime, using relatively elementary tools. The overall strategy—oscillatory integrals, hyperbolic geometry, and the transpose/dual-IFS viewpoint—is attractive and likely to be extendable. The manuscript is largely self-contained and the main structure is clear. However, the central L^2 estimate contains an algebraic error that invalidates Lemma 3.4 as stated; the final theorem may be salvageable, but the proof in its present form does not establish the claimed bound.

major comments (3)
  1. [§3.2, Lemma 3.4, dyadic summation for B_j(a)] The displayed estimate for the non-stationary blocks is algebraically wrong. In the line for B_j(a) the manuscript writes τ^{-3/2}(τ^{-1/2}2^{-j})^{-1} = τ^{-1/2}2^j, but the correct value is τ^{-1}2^j. Consequently the correct aggregate contribution before summing over a is |ξ|^{-1}τ^{2δ−2}, and after summing over a is |ξ|^{-1}τ^{δ−2}. For δ > 1/2 and τ small this term is not controlled by the |ξ|^{-1}τ^{-1} term stated in Lemma 3.4; for instance, taking τ ≍ |ξ|^{-1} gives a contribution ≍ |ξ|^{1−δ}, which exceeds the trivial bound O(1). Thus Lemma 3.4 is false as stated. The theorem may still survive because at τ = |ξ|^{-1/(2δ+1)} the corrected term is dominated by the leading |ξ|^{-1/2}τ^{-1/2}, but the present proof must be repaired by either including the extra term in Lemma 3.4 and redoing the optimization, or by adding a further split using the trivial bound for very small |d_{a'}
  2. [§2.8, Lemmas 2.8 and 2.9] The notation for the reversed word is not typeset correctly: the text writes a := a_n ... a_1, but the intended word must contain inverse letters, and the chain-rule equality γ^T_a(0) = J γ_{a_n ... a_1}(∞) is false without those overlines. The transpose of γ_a is conjugate to γ_a^{-1}, so the correct expression involves γ_{\bar a_n ... \bar a_1}. The estimate |I_{\bar a}| ≍ |I_a| in Lemma 2.8 and the non-concentration counting in Lemma 2.9 both rely on this equality. The intended statement is standard and likely correct, but as printed the proof of Lemma 2.9 is not valid. This is a load-bearing point and must be corrected carefully.
  3. [§3.2, J^+ construction and Claim 1] Several estimates essential to the proof of Lemma 3.4 are asserted without proof: the bound |(φ^-_{a,b})'| ≲ τ^{3/2}, the explicit construction of J^+ and the corresponding φ^+, and the assertions in Claim 1 that |ψ'_{a,b}| ≍ τ and |ψ''_{a,b}| ≲ τ on J^0(a,b). These are used to obtain the |ξ|^{-1}τ^{-1} contribution in Lemma 3.4. In particular, the lower bound |ψ'| ≍ τ in Claim 1 excludes a stationary phase on J^0 and is load-bearing. Please provide the missing computations or a precise derivation for these 'one can check' statements.
minor comments (3)
  1. [§3.1, Lemma 3.1 and its application] Lemma 3.1 assumes a Frostman condition with constant 1, but the Patterson–Sullivan measure only satisfies μ(J) ≤ C r^δ with C = C(Γ). The application should either normalize the measure by C or explicitly track the constant; this is harmless but should be stated.
  2. [§2.8, Lemma 2.8 statement] The definition of \bar a is missing overlines on the letters; it should read \bar a = \bar a_n ... \bar a_1 (or equivalent). This is tied to the major comment above.
  3. [References] Reference [37] lists both an arXiv preprint and an 'Adv. Math. 403 (2022)' citation in the same entry without clear indication of which is final; please clean up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Fourier-decay theorem is derived from external Ahlfors-regularity/non-concentration inputs and oscillatory estimates, with no fitted parameter or load-bearing self-citation.

full rationale

Theorem 1 is obtained by controlling |μ̂(ξ)| via Lemma 3.1 (a Frostman/L2 approximation lemma), Lemma 3.2 (a derivative bound), and the main L2 estimate Lemma 3.4. These lemmas are proved in the paper from bounded distortion, the quasi-Bernoulli property, Ahlfors regularity of the Patterson–Sullivan measure, and the non-concentration Lemma 2.9. The latter is explicitly adapted from Bourgain–Dyatlov [13] by replacing the inverse-image point γ_a^{-1}(∞) with the transpose action γ_a^T(0); the paper does not claim this as a new prediction, and the transpose measure ν is Ahlfors regular by standard PS theory, not by the theorem being proved. Lemma 3.5, the oscillatory integral bound, is attributed to Jordan–Sahlsten [17] and Kaufman [18], with the proof indicated by integration by parts; even though [17] is a self-citation, it is a published, parameter-free external lemma and not an unverified assertion equivalent to the target Fourier decay. No constant is fitted to data, no quantity called a prediction is defined in terms of the Fourier transform, and no uniqueness theorem from the authors is invoked to force the argument. The algebraic issue noted in the proof of Lemma 3.4 is a potential correctness gap, not circularity. Therefore no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters: τ and η are chosen to balance bounds, not fitted to data. The proof relies on established Ahlfors regularity and bounded distortion of PS measures and on standard oscillatory integral estimates.

assumptions (5)
  • domain assumption Schottky disks can be conjugated to avoid 0 (and ∞) in their interiors.
    Section 2.2 'Assumption'; ensures J(x)=-1/x is bilipschitz on all cylinders, used in Lemmas 2.8 and 2.9.
  • domain assumption Bounded distortion and Ahlfors regularity of Patterson-Sullivan measures (cited from [13])
    Lemmas 2.1, 2.2, 2.4, and equation (2); controls cylinder lengths, weights w_{a'}, Frostman exponent, and multiplicity bounds.
  • domain assumption Uniform contraction with exponential factor κ (equation (2), from [13])
    Used in Lemma 2.10 to bound the multiplicity of cylinder covers, which is needed for the non-concentration counting.
  • standard math Oscillatory integral estimate (Lemma 3.5, from [17, Lemma 6.4])
    Van der Corput-type integration-by-parts bound; underpins the I± estimates in the L2 argument.
  • domain assumption The transpose group Γ^T is a Schottky group and its Patterson-Sullivan measure ν=J_*μ is Ahlfors δ-regular
    Section 2.8 and Lemma 2.9; the counting estimate for non-concentration of c/d ratios is the key external input to Lemma 3.8.

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Pith. "Pith review of Quantitative Fourier decay for Patterson-Sullivan measures of dimension larger than $1/2$." pith.science (2026). https://pith.science/paper/IQZ3K4WD

@misc{pith2026260718010,
  author       = {Pith},
  title        = {Pith review of: Quantitative Fourier decay for Patterson-Sullivan measures of dimension larger than $1/2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IQZ3K4WD}},
  note         = {Machine review of arXiv:2607.18010}
}
abstract

We give an elementary proof of power Fourier decay for Patterson-Sullivan measures associated to convex co-compact Schottky groups of dimension $\delta>\frac12$, obtaining the explicit decay exponent $\frac{\delta(2\delta - 1)}{(2\delta + 1)(3 - \delta)}$. The proof replaces the technical machinery used in previous works, such as sum-product estimates, renewal theory, Dolgopyat methods, and $L^2$ flattening, by elementary oscillatory integral estimates, hyperbolic geometry, and a duality argument based on the transpose Schottky group.

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