标题： 支持在部分正规数上的测度 显示英文标题

标题： Measures supported on partly normal numbers

摘要： 一个实数$x$在整数基$b \geq 2$下是正常的，如果其在此基下的数字展开是“公平”的，即对于$k \geq 1$，每个由$k$个来自$\{0, 1, \ldots, b-1\}$的数字组成的有序序列在$x$的数字展开中出现的极限频率相同。 博雷尔的经典结果\cite{b09}断言，勒贝格几乎所有的数$x$在每一个底数$b \geq 2$下都是正规的。 这篇文章分为三部分，考虑了部分正规性的集合。 给定任何整数底数$\mathscr{B}, \mathscr{B}' \subseteq \{2, 3, \ldots\}$的选择，我们研究集合$\mathscr{N}(\mathscr{B}, \mathscr{B}')$的测度论性质，根据定义，该集合的元素在$\mathscr{B}$的底数下是正规的，在$\mathscr{B}'$的底数下是非正规的。 如果任何$(b, b') \in \mathscr{B} \times \mathscr{B}'$是乘法独立的，则一对集合$(\mathscr{B}, \mathscr{B}')$是兼容的。 对于兼容的$(\mathscr{B}, \mathscr{B}')$与$\mathscr{B}' \ne \emptyset$，我们构造在$\mathscr N(\mathscr{B}, \mathscr{B}')$上支持的奇异概率测度，这些测度既是 Frostman 又是 Rajchman，扩展了 Pollington\cite{p81}和 Lyons\cite{l86}的先前工作。 该Rajchman性质完全回答了Kahane和Salem提出的问题\cite{Kahane-Salem-64}，将$\mathscr N(\mathscr{B}, \mathscr{B}')$识别为乘法集（在傅里叶分析的意义上）当且仅当$(\mathscr{B}, \mathscr{B}')$是兼容的。文章的方法论贡献是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整的参数，以确保（子集的）某些性质，如几乎处处正规性、非正规性、球条件和傅里叶衰减。 显示英文摘要

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.