Base-b Expansion Research Articles

Insertion in Constructed Normal Numbers

Abstract Defined by Borel, a real number is normal to an integer base b ≥ 2 if in its base-b expansion every block of digits occurs with the same limiting frequency as every other block of the same length. We consider the problem of insertion in constructed base-b normal expansions to obtain normality to base (b + 1).

Computing absolutely normal numbers in nearly linear time

A real number x is absolutely normal if, for every base b≥2, every two equally long strings of digits appear with equal asymptotic frequency in the base-b expansion of x. This paper presents an explicit algorithm that generates the binary expansion of an absolutely normal number x, with the nth bit of x appearing after npolylog(n) computation steps. This speed is achieved by simultaneously computing and diagonalizing against a martingale that incorporates Lempel-Ziv parsing algorithms in all bases.

Computable irrational numbers with representations of surprising complexity

Cauchy sequences, Dedekind cuts, base-10 expansions and continued fractions are examples of well-known representations of irrational numbers. But there exist others, not so popular, which can be defined using various kinds of sum approximations and best approximations. In this paper we investigate the complexity of a number of such representations.For any fast-growing computable function f, we define an irrational number αf by using a series of reciprocals of powers of all primes. We prove that certain representations of αf are of low computational complexity (which does not depend on f), whereas others, apparently similar representations, can be of arbitrarily high computational complexity (which depends on f). The existence of computable numbers like αf allows us to prove new and non-trivial theorems on the computational complexity of representations without resorting to the standard computability-theoretic machinery involving enumerations and diagonalizations.In the paper we also show how to construct irrational numbers γ whose representations by a Cauchy sequences are of low computational complexity, but whose base-b expansion may be of arbitrarily high computational complexity for all bases b. Moreover, for any E2-irrational number α, there will be an E2-irrational number β, such that α+β has the complexity of γ. As a consequence, two numbers which have, let us say, base-10 expansions of low computational complexity, may add up to a number whose base-10 expansion is of arbitrarily high computational complexity. The same goes for representations by base-2 expansions, base-17 expansions, Dedekind cuts, continued fractions, and so on.

On the computational complexity of algebraic numbers: the Hartmanis–Stearns problem revisited

— We consider the complexity of integer base expansions of algebraic irrational numbers from a computational point of view. We show that the Hartmanis–Stearns problem can be solved in a satisfactory way for the class of multistack machines. In this direction, our main result is that the base-b expansion of an algebraic irrational real number cannot be generated by a deterministic pushdown automaton. We also confirm an old claim of Cobham proving that such numbers cannot be generated by a tag machine with dilation factor larger than one.

A Hot Spot Proof of the Generalized Wall Theorem

If a sequence of numbers behaves like a random sequence, do we expect subsequences to also behave like a random sequence? Wall proved that normality of a base-b expansion is preserved along arithmetic progressions. What makes arithmetic progressions special is that they are deterministic sequences, a type of low-complexity sequence. We give a self-contained proof that selection along deterministic sequences preserves normality and provide several interesting examples of nontrivial deterministic sequences.

On subrecursive representability of irrational numbers, part II

We consider various ways to represent irrational numbers by subrecursive functions. An irrational number can be represented by its base-b expansion; by its base-b sum approximation from below; and by its base-b sum approximation from above. Let S be

Nonnormality of Stoneham constants

This paper examines “Stoneham constants,” namely real numbers of the form \(\alpha_{b,c} = \sum_{n \geq1} 1/(c^{n} b^{c^{n}})\), for coprime integers b≥2 and c≥2. These are of interest because, according to previous studies, α b,c is known to be b-normal, meaning that every m-long string of base-b digits appears in the base-b expansion of the constant with precisely the limiting frequency b −m. So, for example, the constant \(\alpha_{2,3} = \sum_{n \geq1} 1/(3^{n} 2^{3^{n}})\) is 2-normal. More recently it was established that α b,c is not bc-normal, so, for example, α 2,3 is provably not 6-normal. In this paper, we extend these findings by showing that α b,c is not B-normal, where B=b p c q r, for integers b and c as above, p,q,r≥1, neither b nor c divide r, and the condition D=c q/p r 1/p/b c−1<1 is satisfied. It is not known whether or not this is a complete catalog of bases to which α b,c is nonnormal. We also show that the sum of two B-nonnormal Stoneham constants as defined above, subject to some restrictions, is B-nonnormal.

On the expansion of some exponential periods in an integer base

We derive a lower bound for the subword complexity of the base-b expansion (b ≥ 2) of all real numbers whose irrationality exponent is equal to 2. This provides a generalization of a theorem due to Ferenczi and Mauduit. As a consequence, we obtain the first lower bound for the subword complexity of the number e and of some other transcendental exponential periods.

A Nowhere Differentiable Continuous Function

The examples of continuous nowhere differentiable functions given in most analysis texts involve the uniform convergence of a series of functions [1, pp. 401-412]. In the last twenty years this subject has again received a good deal of attention; see [2]-[6]. The purpose of this note is to give a new elementary example that needs only the basic notion of limit and is very accessible and appropriate for a first calculus course. For each real number x E [0, 1], let its base-b expansion be