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Explicit Upper Bounds on Decay Rates of Fourier Transforms of Self-similar Measures on Self-similar Sets

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arxiv 2412.16621 v1 pith:VLHNDHWY submitted 2024-12-21 math.CA math.NT

classification math.CAmath.NT
keywords self-similarsetsdecaymeasuresratesboundsexplicitfourier
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The study of Fourier transforms of probability measures on fractal sets plays an important role in recent research. Faster decay rates are known to yield enhanced results in areas such as metric number theory. This paper focuses on self-similar probability measures defined on self-similar sets. Explicit upper bounds are derived for their decay rates, improving upon prior research. These findings are illustrated with an application to sets of numbers whose digits in their L\"uroth representations are restricted to a finite set.

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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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