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Fourier Decay from $L^2$-Flattening

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arxiv 2407.16699 v3 pith:4G7Y5GQ3 submitted 2024-07-01 math.DS math.CAmath.SP

Simon Baker , Osama Khalil , Tuomas Sahlsten This is my paper · ORCID
classification math.DSmath.CAmath.SP
keywords measuresdecayfouriercocompactconvexflatteningfractalhyperbolic
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abstract

We develop a unified approach for establishing rates of decay for the Fourier transform of a wide class of dynamically defined measures. Among the key features of the method is the systematic use of the $L^2$-flattening theorem obtained in \cite{Khalil-Mixing}, coupled with non-concentration estimates for the derivatives of the underlying dynamical system. This method yields polylogarithmic Fourier decay for Diophantine self-similar measures, and polynomial decay for Patterson-Sullivan measures of convex cocompact hyperbolic manifolds, Gibbs measures associated to non-integrable $C^2$ conformal systems, as well as stationary measures for carpet-like non-conformal iterated function systems. Applications include essential spectral gaps on convex cocompact hyperbolic manifolds, fractal uncertainty principles, and equidistribution properties of typical vectors in fractal sets.

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Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Smooth projections of self-similar measures

    math.DS 2026-07 accept novelty 8.0 of 10

    A spectral-gap criterion gives Sobolev regularity for prescribed projections of self-similar measures and yields explicit examples such as singular measures with all line projections smooth.

  2. $L^2$-Flattening of Self-similar Measures on Non-degenerate Curves

    math.CA 2025-07 conditional novelty 8.0 of 10

    Self-similar measures pushed to non-degenerate curves satisfy optimal L^p-flattening estimates for their Fourier transforms, implying quantitative dimension improvement under convolution.

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

    math.DS 2026-07 conditional novelty 7.0 of 10

    Constructs non-self-similar C∞ fractal systems with non-Rajchman stationary measures and flat-derivative diffeomorphisms that produce polynomial Fourier decay, with a sharp C∞ obstruction for Pisot parameters.

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

    math.DS 2026-07 conditional novelty 7.0 of 10

    Patterson-Sullivan measures of convex co-compact Schottky groups of dimension δ>1/2 satisfy |μ̂(ξ)| ≲ |ξ|^{-δ(2δ-1)/((2δ+1)(3-δ))}.

  5. On the Fourier transform of random Bernoulli convolutions

    math.DS 2025-07 accept novelty 7.0 of 10

    For random Bernoulli convolutions, the Fourier transform is in L^1 almost surely whenever λ_g > 2/π, giving absolute continuity and non-empty interior; polynomial Fourier decay holds for every λ_g.

  6. The curious spectra and dynamics of non-locally finite crystals

    math.SP 2024-11 accept novelty 7.0 of 10

    Periodic graphs with infinitely many neighbours per vertex can have purely singular continuous spectrum, flat bands with only delocalized eigenvectors, and ballistic transport without dispersive estimates.

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