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On oscillatory integrals with H\"older phases

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arxiv 2211.08088 v1 pith:7DUXT3TB submitted 2022-11-15 math.DS math.CA

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keywords arxivmethodolderphenomenonsum-productabsolutelyadditionalallows
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We exhibit a family of autosimilar H\"older maps that satisfies a fractal version of the Van Der Corput Lemma, despite not being absolutely continuous. The result is a direct consequence of a recent work of Sahlsten and Steven arXiv:2009.01703, which is based on a powerful theorem of Bourgain known as a sum-product phenomenon estimate. We give a substantially simpler proof of this fact in our particular context, using an elementary method inspired from arXiv:1704.02909 to check the non-concentration estimates that are needed to apply the sum-product phenomenon. This method allows us to gain additional control over the decay rate.

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  1. 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))}.

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