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Volume 17 (2022): Issue 1 (December 2022)

Volume 16 (2021): Issue 2 (December 2021)

Volume 16 (2021): Issue 1 (June 2021)

Volume 15 (2020): Issue 2 (December 2020)

Volume 15 (2020): Issue 1 (June 2020)

Volume 14 (2019): Issue 2 (December 2019)

Volume 14 (2019): Issue 1 (June 2019)
The sixth International Conference on Uniform Distribution Theory (UDT 2018) CIRM, Luminy, Marseilles, France, October 1–5, 2018

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

Volume 13 (2018): Issue 1 (June 2018)

Volume 12 (2017): Issue 2 (December 2017)

Volume 12 (2017): Issue 1 (June 2017)

Volume 11 (2016): Issue 2 (December 2016)

Volume 11 (2016): Issue 1 (June 2016)

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

Search

Volume 16 (2021): Issue 2 (December 2021)

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

Search

6 Articles
access type Open Access

On the Distribution of αp Modulo One in Quadratic Number Fields

Published Online: 02 Feb 2022
Page range: 1 - 48

Abstract

Abstract

We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [On the distribution of αp modulo one in imaginary quadratic number fields with class number one, J. Théor. Nombres Bordx. 32 (2020), no. 3, 719–760]) and by the first- and second-named authors for real quadratic number fields [Diophantine approximation with prime restriction in real quadratic number fields, Math. Z. (2021)] to 7/22. This generalizes a result of Harman [Diophantine approximation with Gaussian primes, Q. J. Math. 70 (2019), no. 4, 1505–1519] who obtained the same exponent 7/22 for ℚ (i) by extending his method which gave this exponent for ℚ [On the distribution of αp modulo one. II, Proc. London Math. Soc. 72, (1996), no. 3, 241–260]. Our proof is based on an extension of Harman’s sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in the above-mentioned work by the first- and second-named authors, for which we use analytic properties of Hecke L-functions with Größencharacters.

Keywords

  • distribution modulo one
  • Diophantine approximation
  • sieve methods
  • sums over primes
  • quadratic field
  • smoothed sum
  • Poisson summation

MSC 2010

  • Primary 11J17
  • Secondary 11J71
  • 11N35
  • 11N36
  • 11L20
  • 11L07
  • 11K60
access type Open Access

Mahler’s Conjecture on ξ(3/2)nmod 1

Published Online: 02 Feb 2022
Page range: 49 - 70

Abstract

Abstract

K. Mahler’s conjecture: There exists no ξ ∈ ℝ+ such that the fractional parts {ξ(3/2)n} satisfy 0 {ξ(3/2)n} < 1/2 for all n = 0, 1, 2,... Such a ξ, if exists, is called a Mahler’s Z-number. In this paper we prove that if ξ is a Z-number, then the sequence xn = {ξ(3/2)n}, n =1, 2,... has asymptotic distribution function c0(x), where c0(x)=1 for x ∈ (0, 1].

Keywords

  • distribution function
  • fractional part
  • -number

MSC 2010

  • Primary: 11K06
  • Secondary: 11K31
  • 11J71
access type Open Access

Divisibility Parameters and the Degree of Kummer Extensions of Number Fields

Published Online: 02 Feb 2022
Page range: 71 - 88

Abstract

Abstract

Let K be a number field, and let be a prime number. Fix some elements α1,...r of K× which generate a subgroup of K× of rank r. Let n1,...,nr, m be positive integers with mni for every i. We show that there exist computable parametric formulas (involving only a finite case distinction) to express the degree of the Kummer extension K(ζm, α1n1,,αrnr \root {{\ell ^{{n_1}}}} \of {{\alpha _1}} , \ldots ,\root {{\ell ^{{n_r}}}} \of {{\alpha _r}} ) over K(ζm) for all n1,..., nr, m. This is achieved with a new method with respect to a previous work, namely we determine explicit formulas for the divisibility parameters which come into play.

Keywords

  • number field
  • Kummer theory
  • Kummer extension
  • degree

MSC 2010

  • Primary:11Y40
  • Secondary:11R20,11R21
access type Open Access

Balance and Pattern Distribution of Sequences Derived from Pseudorandom Subsets of ℤq

Published Online: 02 Feb 2022
Page range: 89 - 108

Abstract

Abstract

Let q be a positive integer and 𝒮={x0,x1,,xT1}q={0,1,,q1} {\scr S} = \{{x_0},{x_1}, \cdots ,{x_{T - 1}}\}\subseteq {{\rm{\mathbb Z}}_q} = \{0,1, \ldots ,q - 1\} with 0x0<x1<<xT1q1. 0 \le {x_0} < {x_1} <\cdots< {x_{T - 1}} \le q - 1. . We derive from S three (finite) sequences: (1) For an integer M ≥ 2let (sn)be the M-ary sequence defined by sn ≡ xn+1 − xn mod M, n =0, 1,...,T − 2.

(2) For an integer m ≥ 2let (tn) be the binary sequence defined by snxn+1xnmodM,n=0,1,,T2. \matrix{{{s_n} \equiv {x_{n + 1}} - {x_n}\,\bmod \,M,} &#38; {n = 0,1, \cdots ,T - 2.}\cr} n =0, 1,...,T − 2.

(3) Let (un) be the characteristic sequence of S, tn={1if1xn+1xnm1,0,otherwise,n=0,1,,T2. \matrix{{{t_n} = \left\{{\matrix{1 \hfill &#38; {{\rm{if}}\,1 \le {x_{n + 1}} - {x_n} \le m - 1,} \hfill\cr{0,} \hfill &#38; {{\rm{otherwise}},} \hfill\cr}} \right.} &#38; {n = 0,1, \ldots ,T - 2.}\cr} n =0, 1,...,q − 1.

We study the balance and pattern distribution of the sequences (sn), (tn)and (un). For sets S with desirable pseudorandom properties, more precisely, sets with low correlation measures, we show the following:

(1) The sequence (sn) is (asymptotically) balanced and has uniform pattern distribution if T is of smaller order of magnitude than q.

(2) The sequence (tn) is balanced and has uniform pattern distribution if T is approximately un={1ifn𝒮,0,otherwise,n=0,1,,q1. \matrix{{{u_n} = \left\{{\matrix{1 \hfill &#38; {{\rm{if}}\,n \in {\scr S},} \hfill\cr{0,} \hfill &#38; {{\rm{otherwise}},} \hfill\cr}} \right.} &#38; {n = 0,1, \ldots ,q - 1.}\cr} .

(3) The sequence (un) is balanced and has uniform pattern distribution if T is approximately q2.

These results are motivated by earlier results for the sets of quadratic residues and primitive roots modulo a prime. We unify these results and derive many further (asymptotically) balanced sequences with uniform pattern distribution from pseudorandom subsets.

Keywords

  • sequence
  • pseudorandom subset
  • balance
  • pattern distribution
  • correlation measure

MSC 2010

  • 11K45
  • 94A55
  • 11T71
  • 11Z05
access type Open Access

From Randomness in Two Symbols to Randomness in Three Symbols

Published Online: 02 Feb 2022
Page range: 109 - 128

Abstract

Abstract

In 1909 Borel defined normality as a notion of randomness of the digits of the representation of a real number over certain base (fractional expansion). If we think of the representation of a number over a base as an infinite sequence of symbols from a finite alphabet A, we can define normality directly for words of symbols of A: A word x is normal to the alphabet A if every finite block of symbols from A appears with the same asymptotic frequency in x as every other block of the same length. Many examples of normal words have been found since its definition, being Champernowne in 1933 the first to show an explicit and simple instance. Moreover, it has been characterized how we can select subsequences of a normal word x preserving its normality, always leaving the alphabet A fixed. In this work we consider the dual problem which consists of inserting symbols in infinitely many positions of a given word, in such a way that normality is preserved. Specifically, given a symbol b that is not present in the original alphabet A and given a word x that is normal to the alphabet A we solve how to insert the symbol b in infinitely many positions of the word x such that the resulting word is normal to the expanded alphabet A ∪{b}.

Keywords

  • Normal numbers
  • combinatorics on words
  • Champernowne number

MSC 2010

  • Primary 11K16
  • 05A05
access type Open Access

AO. Univ.-Prof. Dr. Reinhard Winkler (1964–2021) An Obituary

Published Online: 02 Feb 2022
Page range: 129 - 136

Abstract

6 Articles
access type Open Access

On the Distribution of αp Modulo One in Quadratic Number Fields

Published Online: 02 Feb 2022
Page range: 1 - 48

Abstract

Abstract

We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [On the distribution of αp modulo one in imaginary quadratic number fields with class number one, J. Théor. Nombres Bordx. 32 (2020), no. 3, 719–760]) and by the first- and second-named authors for real quadratic number fields [Diophantine approximation with prime restriction in real quadratic number fields, Math. Z. (2021)] to 7/22. This generalizes a result of Harman [Diophantine approximation with Gaussian primes, Q. J. Math. 70 (2019), no. 4, 1505–1519] who obtained the same exponent 7/22 for ℚ (i) by extending his method which gave this exponent for ℚ [On the distribution of αp modulo one. II, Proc. London Math. Soc. 72, (1996), no. 3, 241–260]. Our proof is based on an extension of Harman’s sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in the above-mentioned work by the first- and second-named authors, for which we use analytic properties of Hecke L-functions with Größencharacters.

Keywords

  • distribution modulo one
  • Diophantine approximation
  • sieve methods
  • sums over primes
  • quadratic field
  • smoothed sum
  • Poisson summation

MSC 2010

  • Primary 11J17
  • Secondary 11J71
  • 11N35
  • 11N36
  • 11L20
  • 11L07
  • 11K60
access type Open Access

Mahler’s Conjecture on ξ(3/2)nmod 1

Published Online: 02 Feb 2022
Page range: 49 - 70

Abstract

Abstract

K. Mahler’s conjecture: There exists no ξ ∈ ℝ+ such that the fractional parts {ξ(3/2)n} satisfy 0 {ξ(3/2)n} < 1/2 for all n = 0, 1, 2,... Such a ξ, if exists, is called a Mahler’s Z-number. In this paper we prove that if ξ is a Z-number, then the sequence xn = {ξ(3/2)n}, n =1, 2,... has asymptotic distribution function c0(x), where c0(x)=1 for x ∈ (0, 1].

Keywords

  • distribution function
  • fractional part
  • -number

MSC 2010

  • Primary: 11K06
  • Secondary: 11K31
  • 11J71
access type Open Access

Divisibility Parameters and the Degree of Kummer Extensions of Number Fields

Published Online: 02 Feb 2022
Page range: 71 - 88

Abstract

Abstract

Let K be a number field, and let be a prime number. Fix some elements α1,...r of K× which generate a subgroup of K× of rank r. Let n1,...,nr, m be positive integers with mni for every i. We show that there exist computable parametric formulas (involving only a finite case distinction) to express the degree of the Kummer extension K(ζm, α1n1,,αrnr \root {{\ell ^{{n_1}}}} \of {{\alpha _1}} , \ldots ,\root {{\ell ^{{n_r}}}} \of {{\alpha _r}} ) over K(ζm) for all n1,..., nr, m. This is achieved with a new method with respect to a previous work, namely we determine explicit formulas for the divisibility parameters which come into play.

Keywords

  • number field
  • Kummer theory
  • Kummer extension
  • degree

MSC 2010

  • Primary:11Y40
  • Secondary:11R20,11R21
access type Open Access

Balance and Pattern Distribution of Sequences Derived from Pseudorandom Subsets of ℤq

Published Online: 02 Feb 2022
Page range: 89 - 108

Abstract

Abstract

Let q be a positive integer and 𝒮={x0,x1,,xT1}q={0,1,,q1} {\scr S} = \{{x_0},{x_1}, \cdots ,{x_{T - 1}}\}\subseteq {{\rm{\mathbb Z}}_q} = \{0,1, \ldots ,q - 1\} with 0x0<x1<<xT1q1. 0 \le {x_0} < {x_1} <\cdots< {x_{T - 1}} \le q - 1. . We derive from S three (finite) sequences: (1) For an integer M ≥ 2let (sn)be the M-ary sequence defined by sn ≡ xn+1 − xn mod M, n =0, 1,...,T − 2.

(2) For an integer m ≥ 2let (tn) be the binary sequence defined by snxn+1xnmodM,n=0,1,,T2. \matrix{{{s_n} \equiv {x_{n + 1}} - {x_n}\,\bmod \,M,} &#38; {n = 0,1, \cdots ,T - 2.}\cr} n =0, 1,...,T − 2.

(3) Let (un) be the characteristic sequence of S, tn={1if1xn+1xnm1,0,otherwise,n=0,1,,T2. \matrix{{{t_n} = \left\{{\matrix{1 \hfill &#38; {{\rm{if}}\,1 \le {x_{n + 1}} - {x_n} \le m - 1,} \hfill\cr{0,} \hfill &#38; {{\rm{otherwise}},} \hfill\cr}} \right.} &#38; {n = 0,1, \ldots ,T - 2.}\cr} n =0, 1,...,q − 1.

We study the balance and pattern distribution of the sequences (sn), (tn)and (un). For sets S with desirable pseudorandom properties, more precisely, sets with low correlation measures, we show the following:

(1) The sequence (sn) is (asymptotically) balanced and has uniform pattern distribution if T is of smaller order of magnitude than q.

(2) The sequence (tn) is balanced and has uniform pattern distribution if T is approximately un={1ifn𝒮,0,otherwise,n=0,1,,q1. \matrix{{{u_n} = \left\{{\matrix{1 \hfill &#38; {{\rm{if}}\,n \in {\scr S},} \hfill\cr{0,} \hfill &#38; {{\rm{otherwise}},} \hfill\cr}} \right.} &#38; {n = 0,1, \ldots ,q - 1.}\cr} .

(3) The sequence (un) is balanced and has uniform pattern distribution if T is approximately q2.

These results are motivated by earlier results for the sets of quadratic residues and primitive roots modulo a prime. We unify these results and derive many further (asymptotically) balanced sequences with uniform pattern distribution from pseudorandom subsets.

Keywords

  • sequence
  • pseudorandom subset
  • balance
  • pattern distribution
  • correlation measure

MSC 2010

  • 11K45
  • 94A55
  • 11T71
  • 11Z05
access type Open Access

From Randomness in Two Symbols to Randomness in Three Symbols

Published Online: 02 Feb 2022
Page range: 109 - 128

Abstract

Abstract

In 1909 Borel defined normality as a notion of randomness of the digits of the representation of a real number over certain base (fractional expansion). If we think of the representation of a number over a base as an infinite sequence of symbols from a finite alphabet A, we can define normality directly for words of symbols of A: A word x is normal to the alphabet A if every finite block of symbols from A appears with the same asymptotic frequency in x as every other block of the same length. Many examples of normal words have been found since its definition, being Champernowne in 1933 the first to show an explicit and simple instance. Moreover, it has been characterized how we can select subsequences of a normal word x preserving its normality, always leaving the alphabet A fixed. In this work we consider the dual problem which consists of inserting symbols in infinitely many positions of a given word, in such a way that normality is preserved. Specifically, given a symbol b that is not present in the original alphabet A and given a word x that is normal to the alphabet A we solve how to insert the symbol b in infinitely many positions of the word x such that the resulting word is normal to the expanded alphabet A ∪{b}.

Keywords

  • Normal numbers
  • combinatorics on words
  • Champernowne number

MSC 2010

  • Primary 11K16
  • 05A05
access type Open Access

AO. Univ.-Prof. Dr. Reinhard Winkler (1964–2021) An Obituary

Published Online: 02 Feb 2022
Page range: 129 - 136

Abstract

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