论文标题
实际数字在每个基部同样可压缩
Real numbers equally compressible in every base
论文作者
论文摘要
这项工作解决了Lutz和Mayordomo提出的关于不同基础中实数的可压缩性的有限状态可压缩性。 有限状态的可压缩性或等效的有限状态维度可量化无限序列中渐近较低的信息密度。绝对正常的数字，在每个扩展的每个基础上都是不可压缩的，这些数字恰恰是那些具有有限状态尺寸的数字，每个尺寸在每个基数中都等于$ 1 $。例如，在另一个极端情况下，每个理性数字的有限尺寸等于每个基数的$ 0 $。 概括这一点，卢茨和Mayordomo（2021）提出了一个问题：是否存在绝对积极的有限状态维度，严格在0到1之间，等同于0和1-同等地，是否有一个实际数字$ $ξ$和一个$ b $ b $ $ b $ $ b $ $ bepspent-$ $ n $ n $ ac $ spepy的$ s $ s？可以想象没有这样的数字。实际上，有些作品探索了其他可行维度的``零一个''定律 - 即具有某些属性的序列具有可行的维度0或1，在两者之间没有严格的价值。 但是，我们通过证明更普遍的结果来回答卢兹和Mayordomo的问题。我们表明，鉴于任何有理数$ \ langle q_b \ rangle $ $，我们可以明确构建一个数字$ξ$，以便对于任何基本$ b $，base-$ b $ in Base-$ b $ in base-b $ $ b $ is $ b $ is $ q_b $ is $ q_b $。作为一个特殊情况，此结果意味着任何给定的合理尺寸在$ 0 $和$ 1 $之间存在绝对的尺寸数字，如Lutz和Mayordomo所构成的。 在我们的构建中，我们结合了沃尔夫冈·施密特（Wolfgang Schmidt）绝对正常数字的构建（1962）的想法，关于低差异序列的结果以及与指数总和有关的几项新估计。
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 $ξ$ and a compressibility ratio $s \in (0,1)$ such that for every base $b$, the compressibility ratio of the base-$b$ expansion of $ξ$ 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 $ξ$ such that for any base $b$, the finite-state dimension/compression ratio of $ξ$ 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.
