Arbitrarily Fast Quantum Dispersion in Long-Range Crystals

Ga\'etan Leclerc; Mostafa Sabri; Tuomas Sahlsten

REVIEW 2 major objections 4 minor 83 references

Arbitrarily Fast Quantum Dispersion in Long-Range Crystals

T0 review · 2 major / 4 minor · reviewed 2026-08-28 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper constructs long-range crystals whose quantum wave packets disperse at any prescribed polynomial rate, by making the Floquet function a sufficiently lacunary Weierstrass function.

desk verdict A genuinely new result—arbitrarily fast polynomial dispersion via rough Weierstrass Floquet phases—but the printed proof of the main quantitative engine has a notation collision that needs a real fix before the claim is checkable. read the letter →

arxiv 2608.27326 v1 pith:6HO5ND4O submitted 2026-08-27 math.SP math-phmath.CAmath.DSmath.MP

Gaétan Leclerc , Mostafa Sabri , Tuomas Sahlsten This is my paper · ORCID

classification math.SPmath-phmath.CAmath.DSmath.MP MSC 81Q1047B3642A3837C30

keywords Weierstrassfunctionslong-rangecrystalsquantumdispersionFourierdecaytransferoperatorsoscillatoryintegralslocaltimeslacunaryseries

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

The paper claims that a translation-invariant lattice on the integers with hopping only at distances $1,\lambda,\lambda^2,\dots$ and weights $2^{-j}$ spreads localized wave packets so aggressively that the $\ell^1\to\ell^\infty$ norm of the unitary group decays like $|t|^{-N}$ for any prescribed $N$, provided the lacunarity $\lambda$ is large enough. If true, this removes the last apparent speed limit on polynomial dispersion: local finiteness is not needed for fast spreading, and roughness of the phase function that diagonalizes the lattice, far from obstructing decay, is the engine that produces it. Dispersion is reduced to a new Fourier-decay estimate for push-forwards of Lebesgue measure under highly oscillatory Weierstrass phases, with a decay exponent that grows like $1/\alpha$, where $\alpha$ is the Hölder exponent of the phase. The same estimate yields smooth local times for Weierstrass functions of large lacunarity, answering a long-standing question from 1980.

What carries the argument

The load-bearing object is the autosimilarity relation $W(\theta)=\varphi(\theta)+\mu W(\lambda\theta)$, which turns the oscillatory integral into iterates of twisted transfer operators $L_{\lambda,i\Phi}(h)(\theta)=\lambda^{-1}\sum_{a\in A}e^{i\Phi(\theta)}h(g_a(\theta))$, one per digit of the Fourier mode in base $\lambda$. The engine is an $L^2$ contraction estimate for these operators: after $n\asymp \log t/|\log\mu|$ steps the $L^2$ norm of the twisted transfer operator acting on $C^1$ functions is at most $\|h\|_{C^1}t^{-\varepsilon_0}$. Opening the $L^2$ square into pairs of inverse branches and applying an oscillatory-integral step to the phase differences reduces the proof to a tree lemma: the normalized second-derivative sums along branches are uniformly non-concentrated on intervals. For large $\lambda$, a sharper separation lemma yields contraction in a norm adapted to $t$, giving the exponent proportional to $\log\lambda/|\log\mu|$.

What would settle it

Take $\mu=1/2$, $\varphi(\theta)=\cos(2\pi\theta)$, and a fixed moderate $\lambda$ such as $3$, and numerically estimate $\sup_{n}|\int_0^1 e^{itW(\theta)}e^{2\pi in\theta}\,d\theta|$ at large $t$; if the decay exponent is not bounded below by $c(\varphi)\log 3/\log 2$ for the constant $c(\varphi)$ in the large-lacunarity theorem, the bound fails. Equivalently, any explicit interval, branch, and scale $\sigma$ violating the tree-lemma inequality $\lambda^{-n}\#\{a: |\ell_a(\theta)-t(\theta)|<\sigma\}\le \lambda(\sigma^{\gamma/2}+\lambda^{-\gamma n})$ would collapse the proof at that parameter.

Watch

Extended reading notes

Core claim

For every $N\in\mathbb{N}$ there exists a lacunarity threshold $\lambda_N\ge 2$ such that, for every integer $\lambda\ge\lambda_N$, the Hamiltonian $H_w$ with $w_{\pm\lambda^j}=2^{-j}$ and all other weights zero satisfies $\|e^{-itH_w}\|_{\ell^1(\mathbb{Z})\to\ell^\infty(\mathbb{Z})}\lesssim_N |t|^{-N}$ as $|t|\to\infty$. Equivalently, the oscillatory integrals $\sup_{n\in\mathbb{Z}}|\int_0^1 e^{2\pi i n\theta} e^{-itW(\theta)}\,d\theta|$ decay faster than any polynomial. The phase $W$ is the Weierstrass series $W(\theta)=\sum_{j\ge0}\mu^j\varphi(\lambda^j\theta)$ with $\mu=1/2$, $\varphi(\theta)=2\cos(2\pi\theta)$, whose Hölder exponent $\alpha=\log 2/\log\lambda$ tends to $0$ as $\lambda$ grows; the paper proves a Hölder-phase oscillatory integral lemma whose rate improves as $\alpha$ decreases. A companion general result gives the same Fourier decay with a positive exponent for every non-analytic Weierstrass phase, and a matching statement shows exponential dispersion is impossible for any crystal. In the large-lacunarity regime the occupation measure of $W$ has a $C^k$ density, so the spectral measure is purely absolutely continuous.

Load-bearing premise

The proof depends on the tree-lemma assertion that the normalized second-derivative sums along inverse branches never concentrate on short intervals; if this non-concentration failed for any branch, the contraction estimate behind the $|t|^{-N}$ decay would not close.

Editorial extensions

If this is right

For every fixed $N$, a concrete choice of hopping weights gives $\|e^{-itH_w}\|_{\ell^1\to\ell^\infty}\lesssim_N |t|^{-N}$, so the class of long-range crystals has no finite polynomial upper bound on dispersion speed.
Exponential dispersion is impossible for any crystal on $\mathbb{Z}$: the diagonal matrix element cannot decay exponentially, by a compact-support-plus-analyticity argument.
If the hopping weights are eventually monotone, dispersion cannot be faster than $1/t$, so the lacunary arrangement of interactions, not long range itself, is what makes fast dispersion possible.
For sufficiently large $\lambda$, the occupation measure of the Weierstrass phase has a $C^k$ density; in the crystal setting this makes the spectral measure of $\delta_0$ purely absolutely continuous.
When $\alpha=|\log\mu|/\log\lambda<1/2$, the same rough phase produces super-ballistic transport: the position variance diverges at every positive time.

Reading between the lines

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

The transfer-operator mechanism should survive random or multiplicative weights: the paper suggests almost-sure arbitrarily fast polynomial dispersion for random Weierstrass phases with large $\lambda$, but does not prove it.
The large-lacunarity limit identifies the occupation measure with the law of an i.i.d. random series; a natural test is whether Pisot-type arithmetic obstructions (as in Bernoulli convolutions) can make a smoothed version of the phase fail to disperse, which would delimit exactly when roughness helps.
Replacing the Lebesgue source by an arbitrary Frostman measure $\nu$ and asking whether $(W_{\mu,\lambda})_*\nu$ has power Fourier decay for every $\lambda\ge2$ is listed as open; a positive answer would give a genuinely fractal oscillatory integral principle.
In higher dimensions the resonant regions $\{\theta: t\nabla h(\theta)\approx 2\pi k\}$ can have interacting geometry across scales, so the one-dimensional tree lemma does not automatically extend and arbitrarily fast dispersion on $\mathbb{Z}^d$ remains open.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague—this one is worth your time. The paper constructs the first long-range crystals whose dispersion decays like |t|^{-N} for every N, using Weierstrass functions as Floquet phases. It also proves a new C^alpha van der Corput lemma and resolves the Geman–Horowitz local-time question in the large-lacunarity regime. That is actual progress, not a repackaging, and the authors are honest about limits: exponential dispersion is ruled out, monotone weights cannot beat 1/t, and the critical lambda-mu = 1 case remains a conjecture. The proof architecture is sensible: reduce dispersion to Fourier decay of complex push-forwards, exploit autosimilarity through a twisted transfer operator, and control non-concentration with a tree lemma. The separation and tree lemmas (2.3, 2.7–2.9) look internally sound, and the large-lambda bootstrap from L^2 contraction to C^1 contraction in Lemma 2.6 is plausible. The soft spot is real, and it sits in the central quantitative engine. In the proofs of Theorem 2.2 and Theorem 2.10, n is first fixed as n_beta(t) (or n(t) with mu^n t roughly t^{1/2}), and then the same symbol n is “chosen” so that lambda^{2n} is roughly t^{1/8} (or t^{1/(4k)}). Those two choices are incompatible under the stated lambda >= mu^{-10/beta}: with n = n_beta(t), the displayed T is bounded by a negative power of t, not >= t^{1/8}. The stress-test note is right about this. There is also a scale mismatch in psi^{(k)}—the displayed lambda^n should be lambda^{(k+1)n} (lambda^{3n} in the k=2 theorem), and the interval in the tree-lemma step should be correspondingly narrower. These look repairable rather than fatal: the underlying non-concentration estimates are independent of the composition, so a two-count rewrite plus corrected scaling should close the gap. The cross-reference to “Theorem 1.4(3)” and the appendix numbering also need cleanup. I would send this to a serious referee. The novelty is substantial and the flaw is a fixable presentation defect in Section 2, not a hidden counterexample. Once the authors repair the notation and the scaling, I would cite it.

Referee Report

2 major / 4 minor

Summary. This paper studies dispersion (the ℓ1→ℓ∞ decay of the Schrödinger evolution) for translation-invariant long-range Schrödinger operators on Z with lacunary hopping weights. The main result, Theorem 1.1, asserts that for every N there exists λ_N such that, for every integer λ≥λ_N, the Hamiltonian with weights w_{±λ^j}=2^{-j} and all other weights zero satisfies ||e^{-itH_w}||_{ℓ1→ℓ∞} ≲_N |t|^{-N}. The proof is Fourier-analytic: by Floquet theory, dispersion is reduced to uniform-in-m oscillatory integrals against a Weierstrass-type Floquet function, and the main technical contribution is a C^α van der Corput lemma for Weierstrass functions (Theorem 1.4), proved through twisted transfer operators and Dolgopyat-type contraction estimates. The paper also proves that exponential dispersion is impossible for any crystal (Proposition 1.2), gives a local C^α regularity obstruction to fast dispersion (Proposition 3.3), obtains a 1/t obstruction for eventually monotone weights (Corollary 3.6), and derives existence and C^k smoothness of local times for Weierstrass functions of sufficiently large lacunarity (Corollary 1.5), answering a question of Geman and Horowitz in that regime.

Significance. If the main result is correct, it is significant and surprising: it shows that removing local finiteness from a periodic crystal removes the universal polynomial bound that holds in the locally finite one-dimensional setting, and the mechanism is a rough, self-similar Floquet function rather than a smooth one. The paper is self-contained in its central derivation: no parameter is fitted to the dispersive exponent, and the fast-dispersion claim is uniform over all sufficiently large integer λ. The transfer-operator/Dolgopyat approach to Fourier decay of C^α Weierstrass images, and the local-time corollary for large lacunarity, are substantial contributions that go well beyond the specific dispersion application. The obstruction results (Proposition 1.2 and Proposition 3.3) are clean and correctly delimit the possible rates. These strengths justify serious consideration; the issue identified below concerns the checkability of the central quantitative estimate and is, in the referee's view, repairable rather than fatal.

major comments (2)

[§2.3, proof of Theorem 2.10] The proof of Theorem 2.10 uses the same symbol n for two incompatible iteration counts. First n is fixed as n:=n_β(t)=⌊β ln t / |ln μ|⌋, with the standing hypothesis λ≥μ^{-10/β}. A few lines later the text says 'We then choose n such that λ^{2n}≃t^{1/8}' and proceeds with the bounds λ^{-n-2n}≤t^{-2}, T=t μ^{n-1}λ^{-5n-2}≥t^{1/8}, and the subsequent cardinality estimate. These two requirements cannot both hold for a generic λ≥λ_*(β). With n=n_β(t), the hypothesis λ≥μ^{-10/β} gives λ^n≥t^{10} up to constants, so λ^{2n} is at least of order t^{20}, not t^{1/8}; simultaneously T=t μ^{n-1}λ^{-5n-2}≤C t^{-49-β}, so the lower bound T≥t^{1/8} used for the van der Corput step fails. If instead n is chosen so that λ^{2n}≈t^{1/8}, then n is no longer n_β(t), and the identification μ^n≈t^{-β} on which Lemma 2.6 and Theorem 2.5 depend is lost. Since Theorem 2.10 is the engine for the large-λ L^2 contraction used to prove Theorem 1.4(2) and hence Theorem 1.1, the printed proof of the central claim is not checkable as written. A two-count rewrite (for example, an extra count m with λ^{2m}≈t^{1/8} and m≪n_β(t)) may repair the argument, but the scale bookkeeping must be redone.
[§2.2, proof of Theorem 2.2] The same two-n collision occurs in the proof of the fixed-λ L^2-Dolgopyat estimate, Theorem 2.2. After n is chosen through μ^n t ∼ t^{1/2}, the proof again says 'We choose n such that λ^{2n}≃t^{1/(4k)}' and then uses λ^{2n+2n}, g'_{ac}∼λ^{-2n}, and related estimates. As in Theorem 2.10, these two choices of n are not compatible, and this theorem is used for Theorem 1.4(1) and for the general dispersion statement. The issue is not merely cosmetic: the displayed scale T and the error term t^{-1/(4k)} both depend on which count is meant. The proof needs a fresh symbol for the second iteration count and a verification that both required scales can be met simultaneously.

minor comments (4)

[§2.3, proof following Theorem 2.5] The heading 'Proof of Theorem 1.4(3)' should be 'Proof of Theorem 1.4(2)', since the stated Theorem 1.4 has only parts (1) and (2).
[§2.2, Lemma 2.3] In the proof of Lemma 2.3, the text 'This is the statement of Lemma 11.5' should refer to the lemma being proved (Lemma 2.3), not to a nonexistent Lemma 11.5.
[Appendix A, proof of Theorem A.1] The proof of the C^1 van der Corput lemma is headed 'Proof of Theorem A.2', but the statement being proved is Theorem A.1; Theorem A.2 is the later C^{1+α} result.
[§4.1, Corollary 4.4] The phrase 'blows up in finite time' is used without a formal definition; since the preceding paragraph discusses the finiteness of ||X e^{itH} δ_0||, the corollary should state explicitly that this norm is infinite for every t≠0 in the indicated regime.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the arbitrarily fast dispersion theorem is obtained from a self-contained Dolgopyat contraction estimate with no fitted parameters.

full rationale

I walked the derivation chain from Theorem 1.1 through Eq. (1.1), Theorem 1.4, and the Dolgopyat estimates. Theorem 1.1 is not fitted: for fixed μ=1/2 and φ=2cos(2π·), the exponent of Theorem 1.4(2) grows with log λ and the theorem is uniform in all λ above an explicit λ_*(β), so the claim 'for every N there exists λ_N' is a genuine uniform consequence, not a calibration. The quantitative engine (Theorem 2.2, and for the large-λ regime Theorem 2.10) is proved from Lemma 2.3 (separation of derivative sums) and the tree lemmas 2.4/2.9, whose inputs are real-analyticity and non-constancy of φ plus non-analyticity of W; none of these inputs contains the target rate t^{-N}. The phase non-concentration lemmas 2.7–2.9 are proved in the text from sublevel-set estimates for non-constant analytic functions. The only overlapping-author citation in the load-bearing line is Eq. (1.1), attributed to [48, Section 2.2], but that identity is an elementary Floquet diagonalization (Fourier conjugation sends H_w to multiplication by h), is parameter-free, and is not where the rate comes from; the rough regime does not reduce to the [48] results. The smoother regime λμ<1 is credited to external [3] and is not used for the main rough-regime conclusion. Appendix B's self-citations to [52, 81] concern auxiliary Weierstrass regularity facts; the core separation and tree-lemma arguments in Section 2 are self-contained. I therefore find no circular step, no fitted input renamed as a prediction, and no load-bearing self-citation chain.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No free constants are fitted to the target dispersion rate; lambda, mu, and phi are model parameters with theorems uniform over large lambda. The main external input is the Fourier representation from [48] and standard analytic and ergodic tools. No new physical entities are postulated.

free parameters (3)

lacunarity lambda = lambda >= lambda_N; crude bound lambda_N <= 2^{10^9 N}
The model and proof require lambda sufficiently large; the threshold is only shown to exist, with a crude exponential upper bound. It is a construction parameter, not fitted to data.
contraction factor mu = 1/2 in the crystal example
Hopping amplitudes are 2^{-j}; Theorem 1.4 handles any 0 < mu < 1, so this is a model parameter rather than a fitted constant.
base phase phi = 2 cos(2 pi theta) in the crystal example
The theorem is proven for any nonconstant real-analytic 1-periodic phi; the cosine choice realizes the Weierstrass Floquet function for the crystal.

assumptions (5)

domain assumption Dispersion equals uniform Fourier decay of the Floquet push-forward measures (Eq. 1.1, taken from [48]).
This is the bridge from Schrodinger evolution to oscillatory integrals; it is imported from prior work by the authors and is load-bearing for every dispersion statement.
standard math Paley-Wiener theorem and Fourier inversion for compactly supported spectral measures (Proposition 3.1).
Used to rule out exponential dispersion for any crystal.
standard math Classical van der Corput lemma as an input inside the L2-Dolgopyat estimates.
Applied to phases psi_{a,b,c} with controlled second or k-th derivatives.
standard math Aaronson's infinite ergodic theory and Guivarc'h's strict aperiodicity criterion (Appendix B.3).
Used for the critical lacunarity density of partial-sum derivatives, supporting the singular-continuum conjecture rather than the main dispersion theorem.
domain assumption Real analyticity and non-constancy of phi are hypotheses of Theorem 1.4; for the crystal phi = 2 cos is analytic.
The separation and non-concentration lemmas require analyticity; the concrete crystal satisfies it.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Arbitrarily Fast Quantum Dispersion in Long-Range Crystals." pith.science (2026). https://pith.science/paper/6HO5ND4O

@misc{pith2026260827326,
  author       = {Pith},
  title        = {Pith review of: Arbitrarily Fast Quantum Dispersion in Long-Range Crystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6HO5ND4O}},
  note         = {Machine review of arXiv:2608.27326}
}

abstract

We construct the first examples of long-range crystals exhibiting arbitrarily fast polynomial quantum dispersion. The Floquet functions of our Hamiltonians are highly oscillatory Weierstrass functions, whose rough autosimilar structure drives the fast dispersion. The proof develops a new Fourier decay theory for $C^\alpha$ images of Lebesgue measure, based on a Dolgopyat-inspired transfer operator method, and yields a van der Corput lemma for Weierstrass functions. As a consequence, the local time of classical Weierstrass functions of sufficiently large lacunarity exists and is $C^k$, answering a question raised by Geman and Horowitz in 1980.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

83 extracted references · 80 canonical work pages

[1]

Aaronson,An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, Vol

J. Aaronson,An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, Vol. 50, American Mathematical Society, Providence, RI, 1997

1997
[2]

Fourier decay and non-decay for pseudo-affine self-conformal measures

A. Algom, S. Ben Ovadia, F. Rodriguez Hertz and M. Shannon,Fourier decay and non-decay for pseudo- affine self-conformal measures, arXiv:2607.22001, 2026

work page Pith review arXiv 2026
[3]

Algom, Y

A. Algom, Y. Chang, M. Wu and Y.-L. Wu,Van der Corput and metric theorems for geometric progressions for self-similar measures, Math. Ann.393(2025), 183–214

2025
[4]

Algom, F

A. Algom, F. Rodriguez Hertz and Z. Wang,Logarithmic Fourier decay for self-conformal measures, J. Lond. Math. Soc. (2)106(2022), no. 2, 1628–1661

2022
[5]

Polynomial Fourier decay and a cocycle version of Dolgopyat's method for self conformal measures

A. Algom, F. Rodriguez Hertz and Z. Wang,Polynomial Fourier decay and a cocycle version of Dolgopyat’s method for self-conformal measures, arXiv:2306.01275

work page Pith review arXiv
[6]

J. M. Anderson, E. A. Housworth and L. D. Pitt,The spectral theory of multiplication operators and the recurrence properties for nondifferentiable functions in the Zygmund class, Mathematika39(1992), 136–151

work page 1992
[7]

Avila and S

A. Avila and S. Jitomirskaya,Almost localization and almost reducibility, J. Eur. Math. Soc.12(2010), no. 1, 93–131

work page 2010
[8]

Baker and A

S. Baker and A. Banaji,Polynomial Fourier decay for fractal measures and their pushforwards, Math. Ann. 392(2025), no. 1, 209–261

work page 2025

Show all 83 references

[9]

Baker, O

S. Baker, O. Khalil and T. Sahlsten,Fourier decay fromL 2-flattening, arXiv:2407.16699

arXiv
[10]

Baker and T

S. Baker and T. Sahlsten,Spectral gaps and Fourier dimension for self-conformal sets with overlaps, arXiv:2306.01389

arXiv
[11]

Banaji and H

A. Banaji and H. Yu,Fourier transform of nonlinear images of self-similar measures: quantitative aspects, to appear in Peking Math. J., arXiv:2503.07508

arXiv
[12]

Banaji and H

A. Banaji and H. Yu,Fourier transform of nonlinear images of self-similar measures: qualitative aspects, arXiv:2606.09743, 2026

2026 arXiv
[13]

Bambusi and Z

D. Bambusi and Z. Zhao,Dispersive estimate for quasi-periodic Schr¨ odinger operators on 1-d lattices, Adv. Math.366(2020), 107071

2020
[14]

Bara´ nski, B

K. Bara´ nski, B. B´ ar´ any and J. Romanowska,On the dimension of the graph of the classical Weierstrass function, Adv. Math.265(2014), 32–59

2014
[15]

N. K. Bary,A Treatise on Trigonometric Series, Vols. I–II, Pergamon Press, Oxford, 1964

1964
[16]

Beurling and P

A. Beurling and P. Malliavin,On Fourier transforms of measures with compact support, Acta Math.107 (1962), 291–309

1962
[17]

Bourgain and S

J. Bourgain and S. Dyatlov,Fourier dimension and spectral gaps for hyperbolic surfaces, Geom. Funct. Anal. 27(2017), no. 4, 744–771

2017
[18]

Boutet de Monvel and M

A. Boutet de Monvel and M. Sabri,Ballistic transport in periodic and random media, inFrom Complex Analysis to Operator Theory: A Panorama, Oper. Theory Adv. Appl., Vol. 291, Birkh¨ auser/Springer, Cham, 2023, 163–216

2023
[19]

Buczolich,Occupation measure and level sets of the Weierstrass–Cellerier function, inRecent Develop- ments in Fractals and Related Fields, Appl

Z. Buczolich,Occupation measure and level sets of the Weierstrass–Cellerier function, inRecent Develop- ments in Fractals and Related Fields, Appl. Numer. Harmon. Anal., Birkh¨ auser, Boston, 2010, 3–18

2010
[20]

Buczolich, A

Z. Buczolich, A. K¨ aenm¨ aki and B. Maga,Level sets of prevalent Weierstrass functions, arXiv:2507.15591
[21]

M. O. C´ aceres and M. Nizama,The quantum L´ evy walk, J. Phys. A: Math. Theor.43(2010), no. 45, 455306

2010
[22]

C. Chen, B. Li and V. Suomala,Fourier dimension of Mandelbrot multiplicative cascades, Comm. Math. Phys.406(2025), no. 8, 182

2025
[23]

Damanik, J

D. Damanik, J. Fillman and G. Young,Optimal dispersion for discrete periodic Schr¨ odinger operators, arXiv:2505.14475

arXiv
[24]

Damanik and A

D. Damanik and A. Gorodetski,Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian, Comm. Math. Phys.305(2011), no. 1, 221–277

2011
[25]

Damanik, A

D. Damanik, A. Gorodetski and W. Yessen,The Fibonacci Hamiltonian, Invent. Math.206(2016), no. 3, 629–692

2016
[26]

Damanik, M

D. Damanik, M. Luki´ c and W. Yessen,Quantum dynamics of periodic and limit-periodic Jacobi and block Jacobi matrices with applications to some quantum many-body problems, Comm. Math. Phys.337(2015), no. 3, 1535–1561. ARBITRARILY FAST QUANTUM DISPERSION IN LONG-RANGE CRYSTALS 35

2015
[27]

Damanik, T

D. Damanik, T. Malinovitch and G. Young,What is ballistic transport?, J. Spectr. Theory, published online, 2024, DOI: 10.4171/JST/532

2024 doi
[28]

Damanik and S

D. Damanik and S. Tcheremchantsev,Upper bounds in quantum dynamics, J. Amer. Math. Soc.20(2007), no. 3, 799–827

2007
[29]

Dolgopyat,On decay of correlations in Anosov flows, Ann

D. Dolgopyat,On decay of correlations in Anosov flows, Ann. of Math. (2)147(1998), no. 2, 357–390

1998
[30]

Fillman,Ballistic transport for periodic Jacobi operators onZ d, inFrom Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory, Oper

J. Fillman,Ballistic transport for periodic Jacobi operators onZ d, inFrom Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory, Oper. Theory Adv. Appl., Vol. 285, Birkh¨ auser/Springer, Cham, 2021, 57–68

2021
[31]

Firsova,On the time decay of a wave packet in a one-dimensional finite band periodic lattice, J

N. Firsova,On the time decay of a wave packet in a one-dimensional finite band periodic lattice, J. Math. Phys.37(1996), 1171–1181

1996
[32]

W. L. Fouch´ e and S. Mukeru,On local times of Martin-L¨ of random Brownian motion, Theoret. Comput. Sci.979(2023), 114199

2023
[33]

Garban and V

C. Garban and V. Vargas,Harmonic analysis of Gaussian multiplicative chaos on the circle, Probab. Theory Related Fields, published online, 2026, DOI: 10.1007/s00440-026-01497-7

2026 doi
[34]

Geman and J

D. Geman and J. Horowitz,Occupation densities, Ann. Probab.8(1980), no. 1, 1–67

1980
[35]

Guivarc’h,Propri´ et´ es ergodiques, en mesure infinie, de certains syst` emes dynamiques fibr´ es, Ergodic Theory Dynam

Y. Guivarc’h,Propri´ et´ es ergodiques, en mesure infinie, de certains syst` emes dynamiques fibr´ es, Ergodic Theory Dynam. Systems9(1989), no. 3, 433–453

1989
[36]

F. A. Harang and C. Ling,Regularity of local times associated with Volterra–L´ evy processes and path-wise regularization of stochastic differential equations, J. Theoret. Probab.35(2022), 1706–1735

2022
[37]

Haro and J

A. Haro and J. Puig,A Thouless formula and Aubry duality for long-range Schr¨ odinger skew-products, Nonlinearity26(2013), no. 5, 1163–1187

2013
[38]

Hochman,On self-similar sets with overlaps and inverse theorems for entropy, Ann

M. Hochman,On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2)180 (2014), no. 2, 773–822

2014
[39]

B. D. Hughes, M. F. Shlesinger and E. W. Montroll,Random walks with self-similar clusters, Proc. Natl. Acad. Sci. USA78(1981), no. 6, 3287–3291

1981
[40]

B. R. Hunt,The Hausdorff dimension of graphs of Weierstrass functions, Proc. Amer. Math. Soc.126 (1998), no. 3, 791–800

1998
[41]

Imkeller, O

P. Imkeller, O. Menoukeu Pamen, G. dos Reis and A. R´ eveillac,Rough Weierstrass functions and dynamical systems: the smoothness of the SBR measure, Pure Appl. Funct. Anal.7(2022), no. 4, 1273–1305

2022
[42]

Imkeller and O

P. Imkeller and O. Menoukeu Pamen,Takagi type functions and dynamical systems: the smoothness of the SBR measure and the existence and smoothness of local time, arXiv:2107.07185

arXiv
[43]

Jordan and T

T. Jordan and T. Sahlsten,Fourier transforms of Gibbs measures for the Gauss map, Math. Ann.364 (2016), no. 3–4, 983–1023

2016
[44]

Kahane,Some Random Series of Functions, 2nd ed., Cambridge Studies in Advanced Mathematics, Vol

J.-P. Kahane,Some Random Series of Functions, 2nd ed., Cambridge Studies in Advanced Mathematics, Vol. 5, Cambridge University Press, Cambridge, 1985

1985
[45]

Kaufman,Continued fractions and Fourier transforms, Mathematika27(1980), no

R. Kaufman,Continued fractions and Fourier transforms, Mathematika27(1980), no. 2, 262–267

1980
[46]

Kaufman,On Bernoulli convolutions, inConference in Modern Analysis and Probability, Contemp

R. Kaufman,On Bernoulli convolutions, inConference in Modern Analysis and Probability, Contemp. Math., Vol. 26, American Mathematical Society, Providence, RI, 1984, 217–222

1984
[47]

Keller,An elementary proof for the dimension of the graph of the classical Weierstrass function, Ann

G. Keller,An elementary proof for the dimension of the graph of the classical Weierstrass function, Ann. Inst. Henri Poincar´ e Probab. Stat.53(2017), no. 1, 169–181

2017
[48]

Kerner, O

J. Kerner, O. Post, M. Sabri and M. T¨ aufer,The curious spectra and dynamics of non-locally finite crystals, Comm. Math. Phys.406(2025), no. 7, 169

2025
[49]

Killip and B

R. Killip and B. Simon,Sum rules for Jacobi matrices and their applications to spectral theory, Ann. of Math. (2)158(2003), no. 1, 253–321

2003
[50]

J. R. Kinney,Note on a singular function of Minkowski, Proc. Amer. Math. Soc.11(1960), no. 5, 788–794

1960
[51]

Leclerc,On oscillatory integrals with H¨ older phases, Discrete Contin

G. Leclerc,On oscillatory integrals with H¨ older phases, Discrete Contin. Dyn. Syst.44(2024), no. 1, 263–280

2024
[52]

Leclerc,Fourier decay of equilibrium states and the Fibonacci Hamiltonian, arXiv:2507.23731

G. Leclerc,Fourier decay of equilibrium states and the Fibonacci Hamiltonian, arXiv:2507.23731

arXiv
[53]

Leclerc,Fourier decay of equilibrium states for bunched attractors, Nonlinearity37(2024), no

G. Leclerc,Fourier decay of equilibrium states for bunched attractors, Nonlinearity37(2024), no. 9, 095020

2024
[54]

Leclerc,Fourier decay of equilibrium states on hyperbolic surfaces, arXiv:2307.10755

G. Leclerc,Fourier decay of equilibrium states on hyperbolic surfaces, arXiv:2307.10755

arXiv
[55]

Leclerc,Julia sets of hyperbolic rational maps have positive Fourier dimension, Comm

G. Leclerc,Julia sets of hyperbolic rational maps have positive Fourier dimension, Comm. Math. Phys.397 (2023), no. 2, 503–546

2023
[56]

Ledrappier,On the dimension of some graphs, Contemp

F. Ledrappier,On the dimension of some graphs, Contemp. Math.135(1992), 285–293

1992
[57]

Lequen and T

F. Lequen and T. Sahlsten,Quantitative Fourier decay for Patterson–Sullivan measures of dimension larger than1/2, arXiv:2607.18010

arXiv
[58]

Li,Decrease of Fourier coefficients of stationary measures, Math

J. Li,Decrease of Fourier coefficients of stationary measures, Math. Ann.372(2018), no. 3–4, 1189–1238

2018
[59]

Li,Fourier decay, renewal theorem and spectral gaps for random walks on split semisimple Lie groups, Ann

J. Li,Fourier decay, renewal theorem and spectral gaps for random walks on split semisimple Lie groups, Ann. Sci. ´Ec. Norm. Sup´ er. (4)55(2022), no. 6, 1613–1686

2022
[60]

Li and T

J. Li and T. Sahlsten,Fourier transform of self-affine measures, Adv. Math.374(2020), 107349

2020
[61]

Li and T

J. Li and T. Sahlsten,Trigonometric series and self-similar sets, J. Eur. Math. Soc.24(2022), no. 1, 341–368

2022
[62]

Lindenstrauss and P

E. Lindenstrauss and P. P. Varj´ u,Random walks in the group of Euclidean isometries and self-similar measures, Duke Math. J.165(2016), no. 6, 1061–1127. 36 GA ´ETAN LECLERC, MOSTAFA SABRI AND TUOMAS SAHLSTEN

2016
[63]

Mantica and V

G. Mantica and V. Totik,Regularity of Minkowski’s question mark measure, its inverse and a class of IFS invariant measures, J. Lond. Math. Soc. (2)99(2019), 707–732

2019
[64]

Mi and Z

Y. Mi and Z. Zhao,Dispersive estimate for two-periodic discrete one-dimensional Schr¨ odinger operator, J. Math. Anal. Appl.485(2020), no. 1, 123768

2020
[65]

Mi and Z

Y. Mi and Z. Zhao,Dispersive estimates for periodic discrete one-dimensional Schr¨ odinger operators, Proc. Amer. Math. Soc.150(2022), no. 1, 267–277

2022
[66]

C. A. Mosquera and P. S. Shmerkin,Self-similar measures: asymptotic bounds for the dimension and Fourier decay of smooth images, Ann. Acad. Sci. Fenn. Math.43(2018), 823–834

2018
[67]

Naud,Expanding maps on Cantor sets and analytic continuation of zeta functions, Ann

F. Naud,Expanding maps on Cantor sets and analytic continuation of zeta functions, Ann. Sci. ´Ec. Norm. Sup´ er. (4)38(2005), no. 1, 116–153

2005
[68]

Parry and M

W. Parry and M. Pollicott,Zeta functions and the periodic orbit structure of hyperbolic dynamics, Ast´ erisque 187–188(1990)

1990
[69]

Queff´ elec and O

M. Queff´ elec and O. Ramar´ e,Analyse de Fourier des fractions continues ` a quotients restreints, Enseign. Math. (2)49(2003), no. 3–4, 335–356

2003
[70]

Rapaport,On the Rajchman property for self-similar measures onR d, Adv

A. Rapaport,On the Rajchman property for self-similar measures onR d, Adv. Math.403(2022), 108375

2022
[71]

Rezakhanlou,The packing measure of the graphs and level sets of certain continuous functions, Math

F. Rezakhanlou,The packing measure of the graphs and level sets of certain continuous functions, Math. Proc. Cambridge Philos. Soc.104(1988), no. 2, 347–360

1988
[72]

Romanowska,Measure and Hausdorff dimension of randomized Weierstrass-type functions, Nonlinearity 27(2014), no

J. Romanowska,Measure and Hausdorff dimension of randomized Weierstrass-type functions, Nonlinearity 27(2014), no. 4, 787–801

2014
[73]

Sahlsten,Fourier transforms and iterated function systems, inRecent Developments in Fractals and Related Fields, Trends in Mathematics, Vol

T. Sahlsten,Fourier transforms and iterated function systems, inRecent Developments in Fractals and Related Fields, Trends in Mathematics, Vol. F319, Birkh¨ auser, Cham, 2025, 297–346

2025
[74]

Sahlsten and C

T. Sahlsten and C. Stevens,Fourier transform and expanding maps on Cantor sets, Amer. J. Math.146 (2024), no. 4, 945–982

2024
[75]

Shen,Hausdorff dimension of the graphs of the classical Weierstrass functions, Math

W. Shen,Hausdorff dimension of the graphs of the classical Weierstrass functions, Math. Z.289(2018), no. 1–2, 223–266

2018
[76]

Shmerkin,On the exceptional set for absolute continuity of Bernoulli convolutions, Geom

P. Shmerkin,On the exceptional set for absolute continuity of Bernoulli convolutions, Geom. Funct. Anal. 24(2014), no. 3, 946–958

2014
[77]

Simon,Operators with singular continuous spectrum, VII

B. Simon,Operators with singular continuous spectrum, VII. Examples with borderline time decay, Comm. Math. Phys.176(1996), 713–722

1996
[78]

Solomyak,On the random series P ±λn (an Erd˝ os problem), Ann

B. Solomyak,On the random series P ±λn (an Erd˝ os problem), Ann. of Math. (2)142(1995), no. 3, 611–625

1995
[79]

E. M. Stein and R. Shakarchi,Complex Analysis, Princeton Lectures in Analysis, Vol. II, Princeton University Press, Princeton, NJ, 2003

2003
[80]

Tsujii,Fat solenoidal attractors, Nonlinearity14(2001), no

M. Tsujii,Fat solenoidal attractors, Nonlinearity14(2001), no. 5, 1011–1027

2001
[81]

Tsujii and Z

M. Tsujii and Z. Zhang,Smooth mixing Anosov flows in dimension three are exponentially mixing, Ann. of Math. (2)197(2023), no. 1, 65–158

2023
[82]

P. P. Varj´ u,Absolute continuity of Bernoulli convolutions for algebraic parameters, J. Amer. Math. Soc.32 (2019), no. 2, 351–397

2019
[83]

Zygmund,On lacunary trigonometric series, Trans

A. Zygmund,On lacunary trigonometric series, Trans. Amer. Math. Soc.34(1932), 435–446

1932

Pith tools

Reviewed August 28, 2026 · model on record in the stance chip above.

[1] [1]

Aaronson,An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, Vol

J. Aaronson,An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, Vol. 50, American Mathematical Society, Providence, RI, 1997

1997

[2] [2]

Fourier decay and non-decay for pseudo-affine self-conformal measures

A. Algom, S. Ben Ovadia, F. Rodriguez Hertz and M. Shannon,Fourier decay and non-decay for pseudo- affine self-conformal measures, arXiv:2607.22001, 2026

work page Pith review arXiv 2026

[3] [3]

Algom, Y

A. Algom, Y. Chang, M. Wu and Y.-L. Wu,Van der Corput and metric theorems for geometric progressions for self-similar measures, Math. Ann.393(2025), 183–214

2025

[4] [4]

Algom, F

A. Algom, F. Rodriguez Hertz and Z. Wang,Logarithmic Fourier decay for self-conformal measures, J. Lond. Math. Soc. (2)106(2022), no. 2, 1628–1661

2022

[5] [5]

Polynomial Fourier decay and a cocycle version of Dolgopyat's method for self conformal measures

A. Algom, F. Rodriguez Hertz and Z. Wang,Polynomial Fourier decay and a cocycle version of Dolgopyat’s method for self-conformal measures, arXiv:2306.01275

work page Pith review arXiv

[6] [6]

J. M. Anderson, E. A. Housworth and L. D. Pitt,The spectral theory of multiplication operators and the recurrence properties for nondifferentiable functions in the Zygmund class, Mathematika39(1992), 136–151

work page 1992

[7] [7]

Avila and S

A. Avila and S. Jitomirskaya,Almost localization and almost reducibility, J. Eur. Math. Soc.12(2010), no. 1, 93–131

work page 2010

[8] [8]

Baker and A

S. Baker and A. Banaji,Polynomial Fourier decay for fractal measures and their pushforwards, Math. Ann. 392(2025), no. 1, 209–261

work page 2025