Finite-state dimension, introduced early in this century as a finite-state version of classical Hausdorff dimension, is a quantitative measure of the lower asymptotic density of information in an infinite sequence over a finite alphabet, as perceived by finite automata. Finite-state dimension is a robust concept that now has equivalent formulations in terms of finite-state gambling, lossless finite-state data compression, finite-state prediction, entropy rates, and automatic Kolmogorov complexity. The Schnorr-Stimm dichotomy theorem gave the first automata-theoretic characterization of normal sequences, which had been studied in analytic number theory since Borel defined them. This theorem implies that a sequence (or a real number having this sequence as its base-b expansion) is normal if and only if it has finite-state dimension 1. One of the most powerful classical tools for investigating normal numbers is the Weyl criterion, which characterizes normality in terms of exponential sums. Such sums are well studied objects with many connections to other aspects of analytic number theory, and this has made use of Weyl criterion especially fruitful. This raises the question whether Weyl criterion can be generalized from finite-state dimension 1 to arbitrary finite-state dimensions, thereby making it a quantitative tool for studying data compression, prediction, etc. This paper does exactly this. We extend the Weyl criterion from a characterization of sequences with finite-state dimension 1 to a criterion that characterizes every finite-state dimension. This turns out not to be a routine generalization of the original Weyl criterion. Even though exponential sums may diverge for non-normal numbers, finite-state dimension can be characterized in terms of the dimensions of the subsequence limits of the exponential sums. We demonstrate the utility of our criterion though examples.

翻译：本世纪初作为经典Hausdorff维数的有限状态版本引入的有限状态维数，是对有限字母表上无限序列中信息渐近下稠密度的一种定量度量，该度量通过有限自动机来感知。有限状态维数是一个稳健的概念，如今已在有限状态博弈、无损有限状态数据压缩、有限状态预测、熵率以及自动Kolmogorov复杂度等方面具有等价表述。Schnorr-Stimm二分定理首次给出了正规序列的自动机理论刻画——自Borel定义正规序列以来，该序列一直在解析数论中被研究。该定理指出，序列（或以其为基b展开的实数）是正规的当且仅当其有限状态维数为1。研究正规数最强大的经典工具之一是Weyl准则，该准则通过指数和来刻画正规性。这类指数和是经过充分研究的对象，与解析数论的其他方面存在诸多联系，这使得Weyl准则的应用尤为丰硕。由此提出一个问题：Weyl准则能否从有限状态维数1推广到任意有限状态维数，从而成为研究数据压缩、预测等问题的定量工具？本文正实现了这一点。我们将Weyl准则从刻画有限状态维数为1的序列扩展为一个可刻画任意有限状态维数的准则。这并非对原始Weyl准则的常规推广。尽管非正规数的指数和可能发散，但有限状态维数可通过指数和的子序列极限的维数来刻画。我们通过实例展示了该准则的实用性。