支撑在部分正规数上的测度
Measures supported on partly normal numbers
AI总结：
本文研究部分正规数集合的测度论性质，对相容基对构造了支撑于其上的奇异 Frostman-Rajchman 概率测度，并引入可独立调参的偏斜测度以同时控制正规性、非正规性、球条件与 Fourier 衰减。
摘要：
一个实数 $x$ 关于整数基 $b \geq 2$ 是正规的，如果它在该基下的数字展开是“均衡的”，即对于 $k \geq 1$，来自 $\{0, 1, \ldots, b-1\}$ 的每一个长度为 $k$ 的有序数字序列都在 $x$ 的数字展开中以相同的极限频率出现。Borel 的经典结果 \cite{b09} 断言 Lebesgue 几乎每个数 $x$ 在每一个基 $b \geq 2$ 下都是正规的。这篇由三部分组成的文章考虑部分正规性的集合。给定任意整数基的选择 $\mathscr{B}, \mathscr{B}' \subseteq \{2, 3, \ldots\}$，我们研究集合 $\mathscr{N}(\mathscr{B}, \mathscr{B}')$ 的测度论性质，按定义，该集合中的元素在 $\mathscr{B}$ 中的基下正规，而在 $\mathscr{B}'$ 中的基下非正规。如果任意 $(b, b') \in \mathscr{B} \times \mathscr{B}'$ 都是乘法独立的，则称集合对 $(\mathscr{B}, \mathscr{B}')$ 是相容的。对于满足 $\mathscr{B}' \ne \emptyset$ 的相容对 $(\mathscr{B}, \mathscr{B}')$，我们构造支撑在 $\mathscr N(\mathscr{B}, \mathscr{B}')$ 上的奇异概率测度，这些测度同时是 Frostman 测度和 Rajchman 测度，推广了 Pollington \cite{p81} 和 Lyons \cite{l86} 的先前工作。Rajchman 性质完全回答了 Kahane 和 Salem \cite{Kahane-Salem-64} 的一个问题，表明 $\mathscr N(\mathscr{B}, \mathscr{B}')$ 是（在 Fourier 分析意义下的）重数集合当且仅当 $(\mathscr{B}, \mathscr{B}')$ 是相容的。本文的方法论贡献在于构造了一类称为偏斜测度的概率测度。这些测度依赖于若干可以独立调节的参数，以确保（子集具有）诸如几乎处处正规性、非正规性、球条件和 Fourier 衰减等性质。
英文摘要：
A real number $x$ is normal with respect to an integer base $b \geq 2$ if its digit expansion in this base is ``equitable'', in the sense that for $k \geq 1$, every ordered sequence of $k$ digits from $\{0, 1, \ldots, b-1\}$ occurs in the digit expansion of $x$ with the same limiting frequency. Borel's classical result \cite{b09} asserts that Lebesgue-almost every number $x$ is normal in every base $b \geq 2$. This three-part article considers sets of partial normality. Given any choice of integer bases $\mathscr{B}, \mathscr{B}' \subseteq \{2, 3, \ldots\}$, we investigate measure-theoretic properties of the set $\mathscr{N}(\mathscr{B}, \mathscr{B}')$, whose members are, by definition, normal in the bases of $\mathscr{B}$ and non-normal in the bases of $\mathscr{B}'$. A pair of sets $(\mathscr{B}, \mathscr{B}')$ is compatible if any $(b, b') \in \mathscr{B} \times \mathscr{B}'$ is multiplicatively independent. For compatible $(\mathscr{B}, \mathscr{B}')$ with $\mathscr{B}' \ne \emptyset$, we construct singular probability measures supported on $\mathscr N(\mathscr{B}, \mathscr{B}')$ that are both Frostman and Rajchman, extending prior work of Pollington \cite{p81} and Lyons \cite{l86}. The Rajchman property completely answers a question of Kahane and Salem \cite{Kahane-Salem-64}, identifying $\mathscr N(\mathscr{B}, \mathscr{B}')$ as a set of multiplicity (in the Fourier-analytic sense) if and only if $(\mathscr{B}, \mathscr{B}')$ is compatible. The methodological contribution of the article is the construction of a class of probability measures called skewed measures. These measures depend on a number of parameters that can be independently adjusted to ensure (subsets of) properties such as almost everywhere normality, non-normality, ball conditions and Fourier decay.
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