- Journal Details
- Format
- Journal
- eISSN
- 2309-5377
- First Published
- 30 Dec 2013
- Publication timeframe
- 2 times per year
- Languages
- English

#### Search

- Open Access

Distribution Functions for Subsequences of Generalized Van Der Corput Sequences

Page range: 1 - 10

#### Abstract

For an integer b > 1 let (φ_{b}(n))_{n≥0} denote the van der Corput sequence base in b in [0, 1). Answering a question of O. Strauch, C. Aistleitner and M. Hofer showed that the distribution function of (φ_{b}(n), φ_{b}(n + 1), . . . , φ_{b}(n + s − 1))_{n≥0} on [0, 1)^{s} exists and is a copula. The first and third authors of the present paper showed that this phenomenon extends to a broad class of subsequences of the van der Corput sequence. In this result we extend this paper still further and show that this phenomenon is also true for more general numeration systems based on the beta expansion of W. Parry and A. Rényi.

#### Keywords

- Generalised van der Corput sequences
- beta-expansions
- Hartman distributed sequences of integers
- distribution functions

#### MSC 2010

- 11K31
- 40A05

- Open Access

Upper Bounds for Double Exponential Sums Along a Subsequence

Page range: 11 - 24

#### Abstract

We consider a class of double exponential sums studied in a paper of Sinai and Ulcigrai. They proved a linear bound for these sums along the sequence of denominators in the continued fraction expansion of α, provided α is badly-approximable. We provide a proof of a result, which includes a simple proof of their theorem, and which applies for all irrational α.

#### Keywords

- continued fraction
- badly-approximable α
- double-exponential sum
- discrepancy
- Koksma-Hlawka inequality
- Ostrowski expansion

#### MSC 2010

- 11J70
- 11L03
- 11L07

- Open Access

Palindromic Closures and Thue-Morse Substitution for Markoff Numbers

Page range: 25 - 35

#### Abstract

We state a new formula to compute the Markoff numbers using iterated palindromic closure and the Thue-Morse substitution. The main theorem shows that for each Markoff number m, there exists a word v ∈ {a, b}∗ such that m − 2 is equal to the length of the iterated palindromic closure of the iterated antipalindromic closure of the word av. This construction gives a new recursive construction of the Markoff numbers by the lengths of the words involved in the palindromic closure. This construction interpolates between the Fibonacci numbers and the Pell numbers.

#### Keywords

- iterated palindromic closure
- Thue-Morse Substitution
- Markoff spectra

#### MSC 2010

- 68R15
- 52C99

#### Abstract

The LS-sequences are a parametric family of sequences of points in the unit interval. They were introduced by Carbone [4], who also proved that under an appropriate choice of the parameters L and S, such sequences are lowdiscrepancy. The aim of the present paper is to provide explicit constants in the bounds of the discrepancy of LS-sequences. Further, we generalize the construction of Carbone [4] and construct a new class of sequences of points in the unit interval, the generalized LS-sequences.

#### Keywords

- Discrepancy
- LS-sequence
- uniform distribution
- beta-expansion

#### MSC 2010

- 11K38
- 11J71
- 11A67

- Open Access

Uncanny Subsequence Selections That Generate Normal Numbers

Page range: 65 - 75

#### Abstract

Given a real number 0_{.a1a2a3} . . . that is normal to base b, we examine increasing sequences n_{i} so that the number 0_{.an1an2an3} . . . are normal to base b. Classically, it is known that if the n_{i} form an arithmetic progression, then this will work. We give several more constructions including n_{i} that are recursively defined based on the digits a_{i}. Of particular interest, we show that if a number is normal to base b, then removing all the digits from its expansion which equal (b−1) leaves a base-(b−1) expansion that is normal to base (b − 1)

#### Keywords

- normal numbers

#### MSC 2010

- 11K16

- Open Access

On the Closure of the Image of the Generalized Divisor Function

Page range: 77 - 90

#### Abstract

For any real number s, let σ_{s} be the generalized divisor function, i.e., the arithmetic function defined by σ_{s}(n) := ∑_{d|n} d^{s}, for all positive integers n. We prove that for any r > 1 the topological closure of σ−_{r}(N+) is the union of a finite number of pairwise disjoint closed intervals I_{1}, . . . , I_{ℓ}. Moreover, for k = 1, . . . , ℓ, we show that the set of positive integers n such that σ−_{r}(n) ∈ I_{k} has a positive rational asymptotic density d_{k}. In fact, we provide a method to give exact closed form expressions for I_{1}, . . . , I_{ℓ} and d_{1}, . . . , d_{ℓ}, assuming to know r with sufficient precision. As an example, we show that for r = 2 it results ℓ = 3, I_{1} = [1, π^{2}/9], I_{2} = [10/9, π^{2}/8], I_{3} = [5/4, π^{2}/6], d_{1} = 1/3, d2 = 1/6, and d_{3} = 1/2.

#### Keywords

- Arithmetic functions
- sum of divisors
- topological closure
- asymptotic densities

#### MSC 2010

- 11A25
- 11N37
- 11N64
- 11Y99

- Open Access

Notes on the Distribution of Roots Modulo a Prime of a Polynomial

Page range: 91 - 117

#### Abstract

Let f(x) be a monic polynomial in Z[x] with roots α_{1}, . . ., α_{n}. We point out the importance of linear relations among 1, α_{1}, . . . , α_{n} over rationals with respect to the distribution of local roots of f modulo a prime. We formulate it as a conjectural uniform distribution in some sense, which elucidates data in previous papers.

#### Keywords

- distribution
- polynomial
- roots modulo a prime

#### MSC 2010

- 11K

- Open Access

Corrigendum to the h-Critical Number of Finite Abelian Groups

Page range: 119 - 124

#### Abstract

We here correct two errors of our paper cited in the title: one in the statement of Theorem 5 and another in the proof of Theorem 11.

#### Keywords

- critical number
- abelian groups
- sumsets
- restricted sumsets

#### MSC 2010

- Primary 11B75
- Secondary 05D99
- 11B25
- 11P70
- 20K01

- Open Access

A Note on the Continued Fraction of Minkowski

Page range: 125 - 130

#### Abstract

Denote by Θ_{1},Θ_{2}, · · · the sequence of approximation coefficients of Minkowski’s diagonal continued fraction expansion of a real irrational number x. For almost all x this is a uniformly distributed sequence in the interval [0, 1/2 ]. The average distance between two consecutive terms of this sequence and their correlation coefficient are explicitly calculated and it is shown why these two values are close to 1/6 and 0, respectively, the corresponding values for a random sequence in [0, 1/2].

#### Keywords

- Continued fractions
- approximation coefficients
- metrical theory

#### MSC 2010

- 11K50