AI 中文总结
本文为所有$0<s\le1$构造了Fourier维数与Hausdorff维数均为$s$的确定性Cantor集，利用Weil界实现快速Fourier衰减，并证明其自然测度具有指数正交基，首次给出$\mathbb{R}$上奇异Salem谱测度。
AI 中文摘要
对于所有$0< s \le 1$，我们提供了一种新的确定性构造的Cantor集，其Fourier维数和Hausdorff维数均等于$s$。该构造基于一个直接的Cantor-Moran构造，其中收缩比由整数的倒数给出。获得快速Fourier衰减的关键工具是解析数论中的Weil界。此外，我们证明了自然的等权Cantor-Moran测度是期望的测度，具有接近最优的Fourier衰减，并且该测度在其$L^2$空间中允许一个指数正交基$\{e^{2\pi i \lambda x}: \lambda\in \Lambda\}$。这给出了${\mathbb R}^1$中奇异Salem谱测度的第一个例子。
英文摘要
For all $0< s \le 1$, we provide a new deterministic construction of Cantor sets whose Fourier dimension and Hausdorff dimension are both equal to $s$. The construction is based on a straightforward Cantor-Moran construction with contraction ratios given by reciprocals of integers. The key tool to obtain the fast Fourier decay is due to the Weil bound in analytic number theory. Furthermore, we show that the natural equal-weighted Cantor-Moran measure is the desired measure admitting the near optimal Fourier decay and the measure admits an exponential orthonormal basis $\{e^{2πi λx}: λ\in Λ\}$ for its $L^2$ space. This gives the first examples of singular Salem spectral measures in ${\mathbb R}^1$.
