从数字的二进制表示到$A_2$-表示的连续数字投影算子
A Continuous digit projector from binary representations of numbers onto $A_2$-representation
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中文总结 AI 辅助
本文针对拓扑不等价于经典二进制表示的$A_2$-连分数表示，构造了一个连续的数字投影算子，证明其性质并建立其与右移算子、逆算子的关系以确立逆算子的奇异性。
中文摘要 AI 辅助
众所周知，数字的$A_2$-连分数表示在拓扑上不等价于经典二进制表示，因此这类表示的数字投影算子是一个不连续函数。本文引入了一个连续函数，作为将数字的经典二进制表示映射到零冗余$A_2$-连分数表示的数字的数字投影算子的类似物，该函数形式为\\[f(\Delta^2_{\alpha_1\alpha_2...\alpha_{2n-1}\alpha_{2n}...})= \Delta^{A_2}_{(\frac{1}{2})^{1-\alpha_1}(\frac{1}{2})^{\alpha_2}... (\frac{1}{2})^{1-\alpha_{2n-1}}(\frac{1}{2})^{\alpha_{2n}}...}, \alpha_n\in \{0,1\}.\\] 证明了函数$f$是良定的、连续的且单调的。利用数字相对于其二进制表示的正态性质以及勒贝格关于连续单调函数几乎处处存在有限导数的定理，确立了函数$f$的奇异性。本文还建立了所考虑的函数、数字上的右移算子与数字连分数表示的逆算子之间的关系，随后利用该关系确立了逆算子的奇异性。
英文摘要
As is known, the $A_2$-continued representation of numbers is not topologically equivalent to the classical binary representation; therefore, the digit projector of such representations is a discontinuous function. In this paper, we introduce a continuous function that serves as an analogue of the digit projector of the classical binary representation of numbers into the digits of the $A_2$-continued representation with zero redundancy, namely a function of the form \[f(Δ^2_{α_1α_2...α_{2n-1}α_{2n}...})= Δ^{A_2}_{(\frac{1}{2})^{1-α_1}(\frac{1}{2})^{α_2}... (\frac{1}{2})^{1-α_{2n-1}}(\frac{1}{2})^{α_{2n}}...}, α_n\in \{0,1\}.\] It is proved that the function $f$ is well-defined, continuous, and monotone. Using the normal properties of numbers with respect to their binary representation and Lebesgue's theorem asserting the existence of a finite derivative for a continuous monotone function almost everywhere, the singularity of the function $f$ is established. The paper also establishes a relationship between the considered function, the right-shift operator on digits, and the inversor of the continued representation of numbers. This relationship is then used to establish the singularity of the inversor.
