Measuring the complexity of real numbers is of major importance in computer science, for the purpose of knowing which computations are allowed. Consider a non-computable real number $s$, i.e. a real number which cannot be stored on a computer. We can store only an approximation of $x$, for instance by considering a finite bitstring representing a finite prefix of its binary expansion. For a fixed approximation error $\varepsilon>0$, the size of this finite bitstring is dependent on the \textit{algorithmic complexity} of the finite prefixes of the binary expansion of $s$. The \textit{algorithmic complexity} of a binary sequence $x$, often referred to as \textit{Kolmogorov complexity}, is the length of the smallest binary sequence $x'$, for which there exists an algorithm, such that when presented with $x'$ as input, it outputs $x$. The algorithmic complexity of the binary expansion of real numbers is widely studied, but the algorithmic complexity of other ways of representing real numbers remains poorly reported. However, knowing the algorithmic complexity of different representations may allow to define new and more efficient strategies to represent real numbers. Several papers have established an equivalence between the algorithmic complexity of the $q$-ary expansions, with $q \in \mathbb{N}$, $q \geq 2$, i.e. representations of real numbers in any integer base. In this paper, we study the algorithmic complexity of the so-called $\beta$-expansions, which are representations of real numbers in a base $\beta \in (1,2)$ that display a much more complex behavior as compared to the $q$-ary expansion. We show that for a given real number $s$, the binary expansion is a minimizer of algorithmic complexity, and that for every given $\beta \in (1,2)$, there exists a $\beta$-expansion of $s$ which achieves the lower bound of algorithmic complexity displayed by the binary expansion of $s$.

翻译：度量实数的复杂度在计算机科学中具有重要意义，目的是了解哪些计算是可行的。考虑一个不可计算的实数 $s$，即无法存储在计算机上的实数。我们只能存储 $x$ 的近似值，例如通过考虑一个有限比特串来表示其二进制展开的有限前缀。对于固定的近似误差 $\varepsilon>0$，该有限比特串的大小取决于 $s$ 的二进制展开的有限前缀的算法复杂度。二进制序列 $x$ 的算法复杂度，通常称为柯尔莫哥洛夫复杂度，是指存在一个算法，当输入 $x'$ 时输出 $x$ 的最小二进制序列 $x'$ 的长度。实数的二进制展开的算法复杂度已被广泛研究，但其他表示实数方式的算法复杂度却鲜有报道。然而，了解不同表示的算法复杂度可能有助于定义新的、更高效的实数表示策略。已有若干论文建立了 $q$ 进制展开（其中 $q \in \mathbb{N}$，$q \geq 2$，即任意整数基下的实数表示）的算法复杂度之间的等价性。本文研究所谓的 $\beta$-展开的算法复杂度，这种表示在基 $\beta \in (1,2)$ 下具有比 $q$ 进制展开更复杂的行为。我们证明，对于给定的实数 $s$，其二进制展开是算法复杂度的极小值，并且对于每个给定的 $\beta \in (1,2)$，存在 $s$ 的一个 $\beta$-展开，能够达到 $s$ 的二进制展开所显示的算法复杂度下界。