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On the Discrepancy of Two Families of Permuted Van der Corput Sequences 论置换Van der Corput序列的两族差异
Uniform distribution theory Pub Date : 2018-06-01 DOI: 10.1515/udt-2018-0003
Florian Pausinger, Alev Topuzoglu
{"title":"On the Discrepancy of Two Families of Permuted Van der Corput Sequences","authors":"Florian Pausinger, Alev Topuzoglu","doi":"10.1515/udt-2018-0003","DOIUrl":"https://doi.org/10.1515/udt-2018-0003","url":null,"abstract":"Abstract A permuted van der Corput sequence Sbσ $S_b^sigma$ in base b is a one-dimensional, infinite sequence of real numbers in the interval [0, 1), generation of which involves a permutation σ of the set {0, 1,..., b − 1}. These sequences are known to have low discrepancy DN, i.e. t(Sbσ):=lim supN→∞DN(Sbσ)/log N $tleft({S_b^sigma } right): = {rm{lim}},{rm{sup}}_{N to infty } D_N left({S_b^sigma } right)/{rm{log}},N$ is finite. Restricting to prime bases p we present two families of generating permutations. We describe their elements as polynomials over finite fields 𝔽p in an explicit way. We use this characterization to obtain bounds for t(Spσ) $tleft({S_p^sigma } right)$ for permutations σ in these families. We determine the best permutations in our first family and show that all permutations of the second family improve the distribution behavior of classical van der Corput sequences in the sense that t(Spσ)","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"37 1","pages":"47 - 64"},"PeriodicalIF":0.0,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89785308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
Corrigendum to the h-Critical Number of Finite Abelian Groups 有限阿贝尔群的h临界数的勘误
Uniform distribution theory Pub Date : 2017-12-20 DOI: 10.1515/udt-2017-0018
B. Bajnok
{"title":"Corrigendum to the h-Critical Number of Finite Abelian Groups","authors":"B. Bajnok","doi":"10.1515/udt-2017-0018","DOIUrl":"https://doi.org/10.1515/udt-2017-0018","url":null,"abstract":"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.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"4 1","pages":"119 - 124"},"PeriodicalIF":0.0,"publicationDate":"2017-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78075143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Distribution Functions for Subsequences of Generalized Van Der Corput Sequences 广义Van Der Corput序列子序列的分布函数
Uniform distribution theory Pub Date : 2017-12-20 DOI: 10.1515/udt-2017-0011
P. Lertchoosakul, A. Haddley, R. Nair, Michel J. G. Weber
{"title":"Distribution Functions for Subsequences of Generalized Van Der Corput Sequences","authors":"P. Lertchoosakul, A. Haddley, R. Nair, Michel J. G. Weber","doi":"10.1515/udt-2017-0011","DOIUrl":"https://doi.org/10.1515/udt-2017-0011","url":null,"abstract":"Abstract For an integer b > 1 let (φb(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 (φb(n), φb(n + 1), . . . , φb(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.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"3 1","pages":"1 - 10"},"PeriodicalIF":0.0,"publicationDate":"2017-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83725081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Note on the Continued Fraction of Minkowski 关于闵可夫斯基连分数的注记
Uniform distribution theory Pub Date : 2017-12-20 DOI: 10.1515/udt-2017-0019
H. Jager
{"title":"A Note on the Continued Fraction of Minkowski","authors":"H. Jager","doi":"10.1515/udt-2017-0019","DOIUrl":"https://doi.org/10.1515/udt-2017-0019","url":null,"abstract":"Abstract Denote by Θ1,Θ2, · · · 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].","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"6 1","pages":"125 - 130"},"PeriodicalIF":0.0,"publicationDate":"2017-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78062549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Palindromic Closures and Thue-Morse Substitution for Markoff Numbers 马尔可夫数的回文闭包和tue - morse替换
Uniform distribution theory Pub Date : 2017-12-20 DOI: 10.1515/udt-2017-0013
C. Reutenauer, L. Vuillon
{"title":"Palindromic Closures and Thue-Morse Substitution for Markoff Numbers","authors":"C. Reutenauer, L. Vuillon","doi":"10.1515/udt-2017-0013","DOIUrl":"https://doi.org/10.1515/udt-2017-0013","url":null,"abstract":"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.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"236 1","pages":"25 - 35"},"PeriodicalIF":0.0,"publicationDate":"2017-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76633797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Notes on the Distribution of Roots Modulo a Prime of a Polynomial 关于多项式的模a素数的根分布的注意事项
Uniform distribution theory Pub Date : 2017-12-20 DOI: 10.1515/udt-2017-0017
Y. Kitaoka
{"title":"Notes on the Distribution of Roots Modulo a Prime of a Polynomial","authors":"Y. Kitaoka","doi":"10.1515/udt-2017-0017","DOIUrl":"https://doi.org/10.1515/udt-2017-0017","url":null,"abstract":"Abstract Let f(x) be a monic polynomial in Z[x] with roots α1, . . ., αn. We point out the importance of linear relations among 1, α1, . . . , αn 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.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"4 2","pages":"117 - 91"},"PeriodicalIF":0.0,"publicationDate":"2017-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72596509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
From Randomness in Two Symbols to Randomness in Three Symbols 从两个符号的随机性到三个符号的随机性
Uniform distribution theory Pub Date : 2017-11-11 DOI: 10.2478/udt-2021-0010
Ariel Zylber
{"title":"From Randomness in Two Symbols to Randomness in Three Symbols","authors":"Ariel Zylber","doi":"10.2478/udt-2021-0010","DOIUrl":"https://doi.org/10.2478/udt-2021-0010","url":null,"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}.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"28 1","pages":"109 - 128"},"PeriodicalIF":0.0,"publicationDate":"2017-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88766670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Joint Distribution in Residue Classes of the Base-q and Ostrowski Digital Sums 基-q和Ostrowski数字和的剩余类的联合分布
Uniform distribution theory Pub Date : 2017-10-26 DOI: 10.2478/udt-2019-0010
Divyum Sharma
{"title":"Joint Distribution in Residue Classes of the Base-q and Ostrowski Digital Sums","authors":"Divyum Sharma","doi":"10.2478/udt-2019-0010","DOIUrl":"https://doi.org/10.2478/udt-2019-0010","url":null,"abstract":"Abstract Let q be an integer greater than or equal to 2, and let Sq(n)denote the sum of digits of n in base q.For α=[ 0;1,m¯ ],   m≥2, alpha = left[ {0;overline {1,m} } right],,,,m ge 2, let Sα(n) denote the sum of digits in the Ostrowski α-representation of n. Let m1,m2 ≥ 2 be integers with gcd(q-1,m1)=gcd(m,m2)=1 gcd left( {q - 1,{m_1}} right) = gcd left( {m,{m_2}} right) = 1 We prove that there exists δ> 0 such that for all integers r1,r2, | { 0≤n<N:Sq(n)≡r1(mod m1),  Sα(n)≡r2(mod m2) }|=Nm1m2+0(N1-δ). matrix{ {left| {left{ {0 le n < N:{S_q}(n) equiv {r_1}left( {bmod ,{m_1}} right),,,{S_alpha }(n) equiv {r_2}left( {bmod ,{m_2}} right)} right}} right|} cr { = {N over {{m_1}{m_2}}} + 0left( {{N^{1 - delta }}} right).} cr } The asymptotic relation implied by this equality was proved by Coquet, Rhin & Toffin and the equality was proved for the case α=[ 1¯ ] alpha = left[ {bar 1} right] by Spiegelhofer.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"81 1","pages":"1 - 26"},"PeriodicalIF":0.0,"publicationDate":"2017-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79780436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Discrepancy Results for The Van Der Corput Sequence Van Der Corput序列的差异结果
Uniform distribution theory Pub Date : 2017-10-04 DOI: 10.2478/udt-2018-0010
Lukas Spiegelhofer
{"title":"Discrepancy Results for The Van Der Corput Sequence","authors":"Lukas Spiegelhofer","doi":"10.2478/udt-2018-0010","DOIUrl":"https://doi.org/10.2478/udt-2018-0010","url":null,"abstract":"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.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"26 1","pages":"57 - 69"},"PeriodicalIF":0.0,"publicationDate":"2017-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72844737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
p-adic Valuation of Exponential Sums in One Variable Associated to Binomials 二项相关单变量指数和的p进值
Uniform distribution theory Pub Date : 2017-06-27 DOI: 10.1515/udt-2017-0003
F. Castro, R. Figueroa, Puhua Guan
{"title":"p-adic Valuation of Exponential Sums in One Variable Associated to Binomials","authors":"F. Castro, R. Figueroa, Puhua Guan","doi":"10.1515/udt-2017-0003","DOIUrl":"https://doi.org/10.1515/udt-2017-0003","url":null,"abstract":"Abstract In this paper we compute the p-adic valuation of exponential sums associated to binomials F(X) = aXd₁ + bXd₂ over Fp. In particular, its p-adic valuation is constant for a, b ∈ F∗p . As a byproduct of our results, we obtain a lower bound for the sizes of value sets of binomials over Fq.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"29 1","pages":"37 - 53"},"PeriodicalIF":0.0,"publicationDate":"2017-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86672695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
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