On the dimension of orthogonal projections of self-similar measures

IF 1.2 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2025-07-22 DOI:10.1112/jlms.70245

Amir Algom, Pablo Shmerkin

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引用次数: 0

Abstract

Let $ν$ be a self-similar measure on $R^{d}$ , $d ⩾ 2$ , and let $π$ be an orthogonal projection onto a $k$ -dimensional subspace. We formulate a criterion on the action of the group generated by the orthogonal parts of the iterated function system on $π$ , and show that it ensures that the dimension of $π ν$ is preserved; this significantly refines previous results by Hochman–Shmerkin (2012) and Falconer–Jin (2014), and is sharp for projections to lines and hyperplanes. A key ingredient in the proof is an application of a restricted projection theorem of Gan–Guo–Wang (2024).

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关于自相似测度的正交投影的维数

让ν $\nu$是R d $\mathbb {R}^d$上的自相似测量，d小于2 $d\geqslant 2$，设π $\pi$是k $k$维子空间上的正交投影。我们给出了迭代函数系统的正交部分对π $\pi$的作用的判据，并证明了它保证了π ν $\pi \nu$的维数保持不变；这大大改进了Hochman-Shmerkin（2012）和Falconer-Jin（2014）之前的结果，并且对于线和超平面的投影是尖锐的。证明的关键是对Gan-Guo-Wang（2024）的受限投影定理的应用。

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来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.