分形测度的多项式傅里叶衰减及其推进。

IF 1.3 2区 数学 Q1 MATHEMATICS
Mathematische Annalen Pub Date : 2025-01-01 Epub Date: 2025-01-28 DOI:10.1007/s00208-025-03091-z
Simon Baker, Amlan Banaji
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引用次数: 0

摘要

证明了一类非常一般的分形测度μ在一大族非线性映射F: R→R下在R d上的推进表现出多项式傅里叶衰减:对于所有ξ≠0,存在C, η >使得| F μ ^ (ξ) |≤C | ξ | - η。利用此证明了如果Φ = {Φ a:[0,1]→[0,1]}a∈a是由解析压缩组成的迭代函数系统,并且存在a∈a使得Φ a不是仿射映射,则Φ的每一个非原子自共形度量都有多项式傅里叶衰减;这个结果是由Algom, Rodriguez Hertz和Wang同时得到的。我们证明了有关傅立叶唯一性问题、分形不确定性原理、傅立叶限制估计和分形集合中数字的定量等分布性质的应用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Polynomial Fourier decay for fractal measures and their pushforwards.

We prove that the pushforwards of a very general class of fractal measures μ on R d under a large family of non-linear maps F : R d R exhibit polynomial Fourier decay: there exist C , η > 0 such that | F μ ^ ( ξ ) | C | ξ | - η for all ξ 0 . Using this, we prove that if Φ = { φ a : [ 0 , 1 ] [ 0 , 1 ] } a A is an iterated function system consisting of analytic contractions, and there exists a A such that φ a is not an affine map, then every non-atomic self-conformal measure for Φ has polynomial Fourier decay; this result was obtained simultaneously by Algom, Rodriguez Hertz, and Wang. We prove applications related to the Fourier uniqueness problem, Fractal Uncertainty Principles, Fourier restriction estimates, and quantitative equidistribution properties of numbers in fractal sets.

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来源期刊
Mathematische Annalen
Mathematische Annalen 数学-数学
CiteScore
2.90
自引率
7.10%
发文量
181
审稿时长
4-8 weeks
期刊介绍: Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin. The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin. Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.
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