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Uniform distribution theory最新文献

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Notes on the Distribution of Roots Modulo a Prime of a Polynomial III 关于多项式a素模根分布的注解III
Uniform distribution theory Pub Date : 2020-06-01 DOI: 10.2478/udt-2020-0005
Y. Kitaoka
{"title":"Notes on the Distribution of Roots Modulo a Prime of a Polynomial III","authors":"Y. Kitaoka","doi":"10.2478/udt-2020-0005","DOIUrl":"https://doi.org/10.2478/udt-2020-0005","url":null,"abstract":"Abstract Let f (x) bea monicpolynomialwith integer coefficients and integers r1,..., rn with 0 ≤ r1 ≤··· ≤ rn <p the n roots of f (x) ≡ 0mod p for a prime p. We proposed conjectures on the distribution of the point (r1/p,...,rn/p) in the previous papers. One aim of this paper is to revise them for a reducible polynomial f (x), and the other is to show that they imply the one-dimensional equidistribution of r1/p,...,rn/p for an irreducible polynomial f (x) by a geometric way.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"40 1","pages":"104 - 93"},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74264066","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
Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II 数域的Kummer理论与代数数的约化2
Uniform distribution theory Pub Date : 2020-06-01 DOI: 10.2478/udt-2020-0004
Antonella Perucca, Pietro Sgobba
{"title":"Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II","authors":"Antonella Perucca, Pietro Sgobba","doi":"10.2478/udt-2020-0004","DOIUrl":"https://doi.org/10.2478/udt-2020-0004","url":null,"abstract":"Abstract Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓe is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such that the order of (G mod 𝔭) is congruent to a modulo ℓe only depends on a through its ℓ-adic valuation.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"4 1 1","pages":"75 - 92"},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77500663","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
On the (VilB2; α; γ)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System 在(VilB2;α;广义数系统中构造Zaremba-Halton型网的γ)-谐音
Uniform distribution theory Pub Date : 2020-06-01 DOI: 10.2478/udt-2020-0002
V. Ristovska, V. Grozdanov, Tsvetelina Petrova
{"title":"On the (VilB2; α; γ)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System","authors":"V. Ristovska, V. Grozdanov, Tsvetelina Petrova","doi":"10.2478/udt-2020-0002","DOIUrl":"https://doi.org/10.2478/udt-2020-0002","url":null,"abstract":"Abstract In the present paper the so-called (VilBs; α; γ)-diaphony as a quantitative measure for the distribution of sequences and nets is considered. A class of two-dimensional nets ZB2,νκ,μ Z_{{{rm B}_2},nu }^{kappa ,mu } of type of Zaremba-Halton constructed in a generalized B2-adic system or Cantor system is introduced and the (VilB2; α; γ)-diaphony of these nets is studied. The influence of the vector α = (α1, α2) of exponential parameters to the exact order of the (VilB2; α; γ)-diaphony of the nets ZB2,νκ,μ Z_{{{rm B}_2},nu }^{kappa ,mu } is shown. If α1 = α2, then the following holds: if 1 < α2 < 2 the exact order is 𝒪 ( logNN1-ε {{sqrt {log N} } over {{N^{1 - varepsilon }}}} ) for some ε > 0, if α2 = 2 the exact order is 𝒪 ( logNN {{sqrt {log N} } over N} ) and if α2 > 2 the exact order is 𝒪 ( logNN1+ε {{sqrt {log N} } over {{N^{1 + varepsilon }}}} ) for some ε > 0. If α1 > α2, then the following holds: if 1 < α2 < 2 the exact order is 𝒪 ( logNN1-ε {{sqrt {log N} } over {{N^{1 - varepsilon }}}} ) for some ε > 0, if α2 = 2 the exact order is 𝒪 ( 1N {1 over N} ) and if α2 > 2 the exact order is 𝒪 ( logNN1+ε {{sqrt {log N} } over {{N^{1 + varepsilon }}}} ) for some ε > 0. Here N = Bν, where Bν denotes the number of the points of the nets ZB2,νκ,μ Z_{{{rm B}_2},nu }^{kappa ,mu } .","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"27 1","pages":"27 - 50"},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87794735","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
On Proinov’s Lower Bound for the Diaphony 论普罗尼诺夫音阶的下界
Uniform distribution theory Pub Date : 2020-05-31 DOI: 10.2478/udt-2020-0010
Nathan Kirk
{"title":"On Proinov’s Lower Bound for the Diaphony","authors":"Nathan Kirk","doi":"10.2478/udt-2020-0010","DOIUrl":"https://doi.org/10.2478/udt-2020-0010","url":null,"abstract":"Abstract In 1986, Proinov published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the d−dimensional unit cube [Proinov, P. D.:On irregularities of distribution, C. R. Acad. Bulgare Sci. 39 (1986), no. 9, 31–34]. However, his widely cited paper does not contain the proof of this result but simply states that this will appear elsewhere. To the best of our knowledge, this proof was so far only available in a monograph of Proinov written in Bulgarian [Proinov, P. D.: Quantitative Theory of Uniform Distribution and Integral Approximation, University of Plovdiv, Bulgaria (2000)]. The first contribution of our paper is to give a self contained version of Proinov’s proof in English. Along the way, we improve the explicit asymptotic constants implementing recent, and corrected results of [Hinrichs, A.—Markhasin, L.: On lower bounds for the ℒ2-discrepancy, J. Complexity 27 (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: An improved lower bound for the ℒ2-discrepancy, J. Complexity 34 (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. An improved lower bound for the ℒ2-discrepancy, J. Complexity 34 (2016), 68–77].) Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"6 1","pages":"39 - 72"},"PeriodicalIF":0.0,"publicationDate":"2020-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80471821","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: 6
Families of Well Approximable Measures 非常近似测度族
Uniform distribution theory Pub Date : 2020-03-29 DOI: 10.2478/udt-2021-0003
S. Fairchild, Max Goering, Christian Weiss
{"title":"Families of Well Approximable Measures","authors":"S. Fairchild, Max Goering, Christian Weiss","doi":"10.2478/udt-2021-0003","DOIUrl":"https://doi.org/10.2478/udt-2021-0003","url":null,"abstract":"Abstract We provide an algorithm to approximate a finitely supported discrete measure μ by a measure νN corresponding to a set of N points so that the total variation between μ and νN has an upper bound. As a consequence if μ is a (finite or infinitely supported) discrete probability measure on [0, 1]d with a sufficient decay rate on the weights of each point, then μ can be approximated by νN with total variation, and hence star-discrepancy, bounded above by (log N)N−1. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure μ, there exist finite sets whose star-discrepancy with respect to μ is at most (log N)d−12N−1 {left( {log ,N} right)^{d - {1 over 2}}}{N^{ - 1}} . Moreover, we close a gap in the literature for discrepancy in the case d =1 showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"24 1","pages":"53 - 70"},"PeriodicalIF":0.0,"publicationDate":"2020-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75274292","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
Discrete Correlation of Order 2 of Generalized Rudin-Shapiro Sequences on Alphabets of Arbitrary Size 任意大小字母上广义Rudin-Shapiro序列的2阶离散相关
Uniform distribution theory Pub Date : 2020-02-27 DOI: 10.2478/udt-2020-0001
Pierre-Adrien Tahay
{"title":"Discrete Correlation of Order 2 of Generalized Rudin-Shapiro Sequences on Alphabets of Arbitrary Size","authors":"Pierre-Adrien Tahay","doi":"10.2478/udt-2020-0001","DOIUrl":"https://doi.org/10.2478/udt-2020-0001","url":null,"abstract":"Abstract In 2009, Grant, Shallit, and Stoll [Acta Arith. 140 (2009), [345–368] constructed a large family of pseudorandom sequences, called generalized Rudin--Shapiro sequences, for which they established some results about the average of discrete correlation coefficients of order 2 in cases where the size of the alphabet is a prime number or a squarefree product of primes. We establish similar results for an even larger family of pseudorandom sequences, constructed via difference matrices, in the case of an alphabet of any size. The constructions generalize those from Grant et al. In the case where the size of the alphabet is squarefree and where there are at least two prime factors, we obtain an improvement in the error term by comparison with the result of Grant et al.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"33 1","pages":"1 - 26"},"PeriodicalIF":0.0,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89844966","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
Chains of Truncated Beta Distributions and Benford’s Law 截尾Beta分布链与Benford定律
Uniform distribution theory Pub Date : 2019-12-01 DOI: 10.2478/udt-2019-0011
Pongpol Ruankong, S. Sumetkijakan
{"title":"Chains of Truncated Beta Distributions and Benford’s Law","authors":"Pongpol Ruankong, S. Sumetkijakan","doi":"10.2478/udt-2019-0011","DOIUrl":"https://doi.org/10.2478/udt-2019-0011","url":null,"abstract":"Abstract It was proved by Jang et al. that various chains of one-parameter distributions converge to Benford’s law. We study chains of truncated distributions and propose another approach, using a recent convergence result of the Lerch transcendent function, to proving that they converge to Benford’s law for initial Beta distributions with parameters α and 1.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"319 1","pages":"27 - 32"},"PeriodicalIF":0.0,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77459466","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
On Freiman’s 3k − 4 Theorem 关于Freiman的3k−4定理
Uniform distribution theory Pub Date : 2019-12-01 DOI: 10.2478/udt-2019-0013
M. Huicochea
{"title":"On Freiman’s 3k − 4 Theorem","authors":"M. Huicochea","doi":"10.2478/udt-2019-0013","DOIUrl":"https://doi.org/10.2478/udt-2019-0013","url":null,"abstract":"Abstract Let X and Y be nonempty finite subsets of 𝕑 and X +Y its sumset. The structures of X and Y when r(X, Y ):= |X +Y |−|X|−|Y | is small have been widely studied; in particular the Generalized Freiman’s 3k − 4 Theorem describes X and Y when r(X, Y ) ≤ min{|X|, |Y |} − 4. However, not too much is known about X and Y when r(X, Y ) > min{|X|, |Y |} − 4. In this paper we study the structure of X and Y for arbitrary r(X, Y ).","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"63 1","pages":"43 - 68"},"PeriodicalIF":0.0,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87923632","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
Stable Configurations of Repelling Points on Flat Tori 平面环面上排斥点的稳定构型
Uniform distribution theory Pub Date : 2019-12-01 DOI: 10.2478/udt-2019-0016
M. Nechayeva, Burton Randol
{"title":"Stable Configurations of Repelling Points on Flat Tori","authors":"M. Nechayeva, Burton Randol","doi":"10.2478/udt-2019-0016","DOIUrl":"https://doi.org/10.2478/udt-2019-0016","url":null,"abstract":"Abstract Flat tori are analyzed in the context of an intrinsic Fourier-analytic approach to electrostatics on Riemannian manifolds, introduced by one of the authors in 1984 and previously developed for compact hyperbolic manifolds. The approach covers a large class of repelling laws, but does not naturally include laws with singularities at the origin, for which possible accommodations are discussed in the final section of the paper.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"66 1","pages":"102 - 87"},"PeriodicalIF":0.0,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89505415","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
On the Discrepancy of Random Walks on the Circle 关于圆上随机游动的不一致性
Uniform distribution theory Pub Date : 2019-12-01 DOI: 10.2478/udt-2019-0015
A. Bazarova, I. Berkes, M. Raseta
{"title":"On the Discrepancy of Random Walks on the Circle","authors":"A. Bazarova, I. Berkes, M. Raseta","doi":"10.2478/udt-2019-0015","DOIUrl":"https://doi.org/10.2478/udt-2019-0015","url":null,"abstract":"Abstract Let X1,X2,... be i.i.d. absolutely continuous random variables, let Sk=∑j=1kXj {S_k} = sumnolimits_{j = 1}^k {{X_j}} (mod 1) and let D*N denote the star discrepancy of the sequence (Sk)1≤k≤N. We determine the limit distribution of ND N* sqrt N D_N^* and the weak limit of the sequence N(FN(t)-t) sqrt N left( {{F_N}(t) - t} right) in the Skorohod space D[0, 1], where FN (t) denotes the empirical distribution function of the sequence (Sk)1≤k≤N.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"43 1","pages":"73 - 86"},"PeriodicalIF":0.0,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86641884","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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