论文标题
傅立叶维度和线性图案的避免
Fourier dimension and avoidance of linear patterns
论文作者
论文摘要
本文的结果有两种类型。一方面，我们构建了一组大傅里叶维度，避免了某些类别的线性方程的非平凡解。特别是，给定的任何有限集合的translation-invariant线性方程的任何有限集合\ begin {equination} \ sum_ {i = 1}^v m_ix_i = m_0x_0，\; \ text {with}（m_0，m_1，\ cdots，m_v）\ in \ mathbb n^{v+1}，m_0 = \ sum_ {i = 1}^{v} m_i \ text {和text {and} \ subSeteq [0,1]尺寸1的$，其中不包含任何这些方程的非平凡解；换句话说，在e^{v+1} $中，不存在vector $（x_0，x_1，\ cdots，x_v）\，带有满足任何给定方程的不同条目。这种结构的变体也可以用于获取塞勒姆集合，以避免其他类型的翻译不变线性方程的解决方案，例如，当避免使用的线性方程的集合是不可数量的或具有不合理系数的情况时。尽管这种结构似乎表明塞勒姆集可以避免许多配置，但我们的第二种结果提供了对立面。我们表明，$ \ Mathbb r $的集合超过$ 2/（v+1）$，无法避免上述所有方程式的非平凡解决方案。特别是，一组积极的傅立叶维度必须包含上述形式的非平凡线性模式，以便某些$ v $，因此不能合理独立。这与已知的结果\ cite {m17}形成鲜明对比，后者确保存在完整的Hausdorff尺寸的合理独立集。后一类结果可以被视为塞勒姆正面尺寸的结构丰富度的定量证据，即使维度很小。
The results in this paper are of two types. On one hand, we construct sets of large Fourier dimension that avoid nontrivial solutions of certain classes of linear equations. In particular, given any finite collection of translation-invariant linear equations of the form \begin{equation} \sum_{i=1}^v m_ix_i=m_0x_0, \; \text{ with } (m_0, m_1, \cdots, m_v) \in \mathbb N^{v+1}, m_0 = \sum_{i=1}^{v} m_i \text{ and } v \geq 2, \label{rational-eqn} \end{equation} we find a Salem set $E \subseteq [0,1]$ of dimension 1 that contains no nontrivial solution of any of these equations; in other words, there does not exist a vector $(x_0, x_1, \cdots, x_v) \in E^{v+1}$ with distinct entries that satisfies any of the given equations. Variants of this construction can also be used to obtain Salem sets that avoid solutions of translation-invariant linear equations of other kinds, for instance, when the collection of linear equations to be avoided is uncountable or has irrational coefficients. While such constructions seem to suggest that Salem sets can avoid many configurations, our second type of results offers a counterpoint. We show that a set in $\mathbb R$ whose Fourier dimension exceeds $2/(v+1)$ cannot avoid nontrivial solutions of all equations of the above form. In particular, a set of positive Fourier dimension must contain a nontrivial linear pattern of the above form for some $v$, and hence cannot be rationally independent. This is in stark contrast with known results \cite{M17} that ensure the existence of rationally independent sets of full Hausdorff dimension. The latter class of results may be viewed as quantitative evidence of the structural richness of Salem sets of positive dimension, even if the dimension is arbitrarily small.
