Finite-State Dimension and Real Arithmetic

Doty, David; Lutz, Jack H.; Nandakumar, Satyadev

doi:10.1007/11786986_47

Cited by 12 publications

(10 citation statements)

References 29 publications

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“…D. Doty, J. H. Lutz, and S. Nandakumar took a substantially different approach from D. D. Wall and strengthened his result. They proved in [Doty et al, 2007] that for every real number and every non-zero rational number the -ary expansions of , ( ), and ( ) all have the same finite-state dimension and the same finite-state strong dimension. It follows that and preserve -normality.…”

Section: Normality Preserving Functionsmentioning

confidence: 99%

Some complexity results in the theory of normal numbers

Airey¹,

Jackson

Mance³

2020

Can. J. Math.-J. Can. Math.

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Let N () be the set of real numbers which are normal to base. A well-known result of H. Ki and T. Linton [Ki and Linton, 1994] is that N () is 0 3-complete. We show that the set N ⊥ () of reals which preserve N () under addition is also 0 3-complete. We use the characterization of N ⊥ () given by G. Rauzy in terms of an entropy-like quantity called the noise. It follows from our results that no further characterization theorems could result in a still better bound on the complexity of N ⊥ (). We compute the exact descriptive complexity of other naturally occurring sets associated with noise. One of these is complete at the 0 4 level. Finally, we get upper and lower bounds on the Hausdorff dimension of the level sets associated with the noise.

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Section: Normality Preserving Functionsmentioning

confidence: 99%

Some complexity results in the theory of normal numbers

Airey¹,

Jackson

Mance³

2020

Can. J. Math.-J. Can. Math.

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“…The question of how the complexity or measure theoretic properties of a real number are altered when it is transformed via a real-valued function goes back at least to Wall [14], who showed that adding or multiplying a nonzero rational number to a real number whose base-k expansion is normal 1 yields another real with a normal base-k expansion. Doty, Lutz, & Nandakumar recently extended Wall's result, showing that the finite-state dimension of the base-k expansion of a real number is preserved under addition or multiplication by a nonzero rational number [4]. At the other extreme of the complexity spectrum, it is not hard to show that algorithmic randomness (Martin-Löf randomness [13]) is preserved under addition or multiplication by a nonzero computable real, regardless of the base of the expansion.…”

Section: Introductionmentioning

confidence: 96%

Functions that preserve p-randomness

Fenner

2013

Information and Computation

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We show that polynomial-time randomness (p-randomness) is preserved under a variety of familiar operations, including addition and multiplication by a nonzero polynomial-time computable real number. These results follow from a general theorem: If I ⊆ R is an open interval, f : I → R is a function, and r ∈ I is p-random, then f (r) is p-random provided 1. f is p-computable on the dyadic rational points in I, and 2. f varies sufficiently at r, i.e., there exists a real constant C > 0 such that eitherOur theorem implies in particular that any analytic function about a p-computable point whose power series has uniformly p-computable coefficients preserves p-randomness in its open interval of absolute convergence. Such functions include all the familiar functions from first-year calculus.

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“…This formulation, initially motivated by practical constraints, has proved to be rich and mathematically robust, having several equivalent characterizations. It has unexpected connections to areas such as number theory, information theory, and convex analysis [KN74], [DLN06]. Schnorr and Stimm [SS72] established a particularly significant connection by showing that a number is Borel normal in base b (see for example, [Niv56]) if and only if its base b expansion has finite-state compressibility equal to 1, i.e., is incompressible (see also: [BH13], [BHV05]).…”

Section: Introductionmentioning

confidence: 99%

A Weyl Criterion for Finite-State Dimension and Applications

Lutz¹,

Nandakumar²,

Pulari³

2021

Preprint

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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.

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Finite-State Dimension and Real Arithmetic

Cited by 12 publications

References 29 publications

Some complexity results in the theory of normal numbers

Some complexity results in the theory of normal numbers

Functions that preserve p-randomness

A Weyl Criterion for Finite-State Dimension and Applications

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