Long progressions in sets of fractional dimension
Long progressions in sets of fractional dimension
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发表时间：
2013-08
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通讯作者：
M. Carnovale
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作者：
M. Carnovale
在“高阶傅立叶维数”足够接近于1的实线的某些子集中，我们证明了k+1项等差数列。这个傅立叶维数，在之前的工作中介绍过，是几何测量理论的傅立叶维数的高阶（在加性组合学和均匀性范数的意义上）扩展，可以理解为要求一个测量的均匀性范数，限制在给定的尺度，随着尺度的增加而衰减。我们进一步得到了所考虑的$\mathbb{R}$的子集中包含的算术级数的公共距离集的大小和$L^p$正则性的定量信息。
We demonstrate $k+1$-term arithmetic progressions in certain subsets of the real line whose "higher-order Fourier dimension" is sufficiently close to 1. This Fourier dimension, introduced in previous work, is a higher-order (in the sense of Additive Combinatorics and uniformity norms) extension of the Fourier dimension of Geometric Measure Theory, and can be understood as asking that the uniformity norm of a measure, restricted to a given scale, decay as the scale increases. We further obtain quantitative information about the size and $L^p$ regularity of the set of common distances of the artihmetic progressions contained in the subsets of $\mathbb{R}$ under consideration.