# Volume 13 (2018): Issue 2 (December 2018)

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

#### Search

7 Articles Open Access

#### On Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, II (Constructive Bounds)

Published Online: 25 Jan 2019
Page range: 1 - 21

#### Abstract

In Part I of this paper we studied the irregularities of distribution of binary sequences relative to short arithmetic progressions. First we introduced a quantitative measure for this property. Then we studied the typical and minimal values of this measure for binary sequences of a given length. In this paper our goal is to give constructive bounds for these minimal values.

#### Keywords

• arithmetic progressions
• irregularities of distribution
• binary sequences

#### MSC 2010

• Primary 11K38
• Secondary 11B25 Open Access

#### Optimal Quantization for Piecewise Uniform Distributions

Published Online: 25 Jan 2019
Page range: 23 - 55

#### Abstract

Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite number of points. In this paper, firstly a general approach to this process is outlined using independent random variables and ergodic maps; these give asymptotically the optimal sets of n-means and the nth quantization errors for all positive integers n. Secondly two piecewise uniform distributions are considered on R: one with infinite number of pieces and one with finite number of pieces. For these two probability measures, we describe the optimal sets of n-means and the nth quantization errors for all n ∈ N. It is seen that for a uniform distribution with infinite number of pieces to determine the optimal sets of n-means for n ≥ 2 one needs to know an optimal set of (n − 1)-means, but for a uniform distribution with finite number of pieces one can directly determine the optimal sets of n-means and the nth quantization errors for all n ∈ N.

#### Keywords

• Optimal quantizers
• quantization error
• uniform distribution

#### MSC 2010

• 60Exx
• 94A34 Open Access

#### Discrepancy Results for The Van Der Corput Sequence

Published Online: 25 Jan 2019
Page range: 57 - 69

#### Abstract

Let dN = NDN (ω) be the discrepancy of the van der Corput sequence in base 2. We improve on the known bounds for the number of indices N such that dN log N/100. Moreover, we show that the summatory function of dN satisfies an exact formula involving a 1-periodic, continuous function. Finally, we give a new proof of the fact that dN is invariant under digit reversal in base 2.

#### Keywords

• van der Corput sequence
• irregularities of distribution
• digit reversal

#### MSC 2010

• Primary: 11K38, 11A63
• Secondary: 11K31 Open Access

#### Sets of Bounded Remainder for The Billiard on A Square

Published Online: 25 Jan 2019
Page range: 71 - 82

#### Abstract

We study sets of bounded remainder for the billiard on the unit square. In particular, we note that every convex set S whose boundary is twice continuously differentiable with positive curvature at every point, is a bounded remainder set for almost all starting angles a and every starting point x. We show that this assertion for a large class of sets does not hold for all irrational starting angles α.

#### Keywords

• Bounded remainder set
• billiard path
• discrepancy
• distribution modulo 1
• unfolding-technique

#### MSC 2010

• 11K31
• 11K38 Open Access

#### On The Classification of Ls-Sequences

Published Online: 25 Jan 2019
Page range: 83 - 92

#### Abstract

This paper addresses the question whether the LS-sequences constructed in [Car12] yield indeed a new family of low-discrepancy sequences. While it is well known that the case S = 0 corresponds to van der Corput sequences, we prove here that the case S = 1 can be traced back to symmetrized Kronecker sequences and moreover that for S ≥ 2 none of these two types occurs anymore. In addition, our approach allows for an improved discrepancy bound for S = 1 and L arbitrary.

#### Keywords

• Low-discrepancy
• LS-sequences
• Kronecker-sequences
• classification
• uniform distribution

#### MSC 2010

• 11K38
• 11K31
• 11K06 Open Access

#### Log-Like Functions and Uniform Distribution Modulo One

Published Online: 25 Jan 2019
Page range: 93 - 101

#### Abstract

For a function f satisfying f (x) = o((log x) K), K > 0, and a sequence of numbers (qn) n, we prove by assuming several conditions on f that the sequence (αf (qn)) n≥n0 is uniformly distributed modulo one for any nonzero real number α. This generalises some former results due to Too, Goto and Kano where instead of (qn) n the sequence of primes was considered.

#### Keywords

• Uniform Distribution
• Discrepancy

#### MSC 2010

• 11J71
• 11K38 Open Access

#### On M. B. Levin’s Proofs for The Exact Lower Discrepancy Bounds of Special Sequences and Point Sets (A Survey)

Published Online: 25 Jan 2019
Page range: 103 - 130

#### Abstract

The goal of this overview article is to give a tangible presentation of the breakthrough works in discrepancy theory [3, 5] by M. B. Levin. These works provide proofs for the exact lower discrepancy bounds of Halton’s sequence and a certain class of (t, s)-sequences. Our survey aims at highlighting the major ideas of the proofs and we discuss further implications of the employed methods. Moreover, we derive extensions of Levin’s results.

#### Keywords

• Discrepancy
• Lower bounds
• special sequences

• 11K06
• 11K38