This work solves an open question in finite-state compressibility posed by Lutz and Mayordomo about compressibility of real numbers in different bases. Finite-state compressibility, or equivalently, finite-state dimension, quantifies the asymptotic lower density of information in an infinite sequence. Absolutely normal numbers, being finite-state incompressible in every base of expansion, are precisely those numbers which have finite-state dimension equal to $1$ in every base. At the other extreme, for example, every rational number has finite-state dimension equal to $0$ in every base. Generalizing this, Lutz and Mayordomo (2021) posed the question: are there numbers which have absolute positive finite-state dimension strictly between 0 and 1 - equivalently, is there a real number $\xi$ and a compressibility ratio $s \in (0,1)$ such that for every base $b$, the compressibility ratio of the base-$b$ expansion of $\xi$ is precisely $s$? It is conceivable that there is no such number. Indeed, some works explore ``zero-one'' laws for other feasible dimensions - i.e. sequences with certain properties either have feasible dimension 0 or 1, taking no value strictly in between. However, we answer the question of Lutz and Mayordomo affirmatively by proving a more general result. We show that given any sequence of rational numbers $\langle q_b \rangle$, we can explicitly construct a single number $\xi$ such that for any base $b$, the finite-state dimension/compression ratio of $\xi$ in base-$b$ is $q_b$. As a special case, this result implies the existence of absolutely dimensioned numbers for any given rational dimension between $0$ and $1$, as posed by Lutz and Mayordomo. In our construction, we combine ideas from Wolfgang Schmidt's construction of absolutely normal numbers (1962), results regarding low discrepancy sequences and several new estimates related to exponential sums.
翻译:这项工作解决了由Lutz 和 Mayodomo 提出的有关在不同基数中真实数字压缩的有限状态压缩的开放问题。 Lutz 和 Mayodomo (2021年) 概括了这个问题: 绝对的固定状态压缩, 或相当的有限状态压缩, 在无限的序列中量化了信息无症状的低密度。 绝对的正常数字, 在每个基数中均不可压缩, 就是每个基数的有限状态维度等于$1美元。 在另一个极端, 每个基数的固定状态内, 每个基数的固定状态内, 每个基数的固定度内值等于 $0美元。 我们的正常状态内值内值和内值之间, 一定的正常状态内值内值内值 。 我们的正常状态内值和内值内值内值内值之间, 一定的内值内值内值内, 我们的内值内值或内值内值内值内值内, 我们的内值内值内数, 或内数内数内数内, 我们的内数内数内, 直为 基数。 基数 基数 基数 底数 直的内, 直为 底值内数 底数 直为 直为 。 。