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, finite-state data compression, finite-state prediction, entropy rates, and automatic Kolmogorov complexity. The 1972 Schnorr-Stimm dichotomy theorem gave the first automatatheoretic characterization of normal sequences, which had been studied in analytic number theory since Borel defined them in 1909. This theorem implies, in present-day terminology, 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 1916 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 the Weyl criterion especially fruitful. This raises the question whether the 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. In case the exponential sums are convergent, they converge to the Fourier coefficients of a probability measure whose dimension is precisely the finite state dimension of the sequence. We demonstrate the utility of this formulation through applications, including a new, Fourier analytic, proof of Schnorr and Stimm's landmark result.

In the study of normal numbers using the base-b representation, b 2, we know that a number is normal if and only if it is simply normal in all the bases b n , n 1. We prove the analogue of this result for continued fraction normality. In particular, we show that two notions of continued fraction normality, one where overlapping occurrences of finite patterns are counted as distinct occurrences, and another where only disjoint occurrences are counted as distinct, are identical. This equivalence involves an analogue of a theorem due to Pillai (Proc. Indian Acad. Sci. Sect. A 11 (1940) 73-80) for base-b expansions. The proof requires techniques which are fundamentally different, since the continued fraction expansion utilizes a countably infinite alphabet, leading to a non-compact space.Utilizing the equivalence of these two notions, we provide a new proof of Heersink and Vandehey's recent result that selection of subsequences along arithmetic progressions does not preserve continued fraction normality (Heersink and Vandehey, Arch. Math. 106 (2016) 363-370).

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