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Les Huit Premiers Travaux de Pierre Liardet Pierre Liardet的前八件作品
Uniform distribution theory Pub Date : 2016-12-01 DOI: 10.1515/udt-2016-0019
M. Waldschmidt
{"title":"Les Huit Premiers Travaux de Pierre Liardet","authors":"M. Waldschmidt","doi":"10.1515/udt-2016-0019","DOIUrl":"https://doi.org/10.1515/udt-2016-0019","url":null,"abstract":"Abstract Ce texte est une présentation résumée des huit premiers travaux de Pierre Liardet. Il reprend l’exposé donné à l’Université de Savoie Mont Blanc (Le Bourget-du-Lac) lors du colloque Théorie des Nombres, Systèmes de Numération, Théorie Ergodique les 28 et 29 septembre 2015, un colloque inspiré par les mathématiques de Pierre Liardet. Le premier texte publié par Pierre Liardet l’a été en 1969 dans les Comptes Rendus de l’Académie des Sciences de Paris, il est intitulé “Transformations rationnelles laissant stables certains ensembles de nombres algébriques”, avec Madeleine Ventadoux comme coauteur. Ils étendent des résultats de Gérard Rauzy. Dans la lignée de ces premiers travaux, il s’est attaqué à une conjecture de Władysław Narkiewicz sur les transformations polynomiales et rationnelles. En 1976, avec Ken K. Kubota, il a finalement réfuté cette conjecture. Il a ensuite obtenu des résultats précurseurs sur une conjecture de Serge Lang, qui sont très souvent cités. Nous donnerons un bref survol des résultats qui ont suivi cette percée significative.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"24 1","pages":"169 - 177"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87934475","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
Spatial Equidistribution of Binomial Coefficients Modulo Prime Powers 模素数幂的二项式系数的空间均匀分布
Uniform distribution theory Pub Date : 2016-12-01 DOI: 10.1515/udt-2016-0017
G. Barat, P. Grabner
{"title":"Spatial Equidistribution of Binomial Coefficients Modulo Prime Powers","authors":"G. Barat, P. Grabner","doi":"10.1515/udt-2016-0017","DOIUrl":"https://doi.org/10.1515/udt-2016-0017","url":null,"abstract":"Abstract The spatial distribution of binomial coefficients in residue classes modulo prime powers is studied. It is proved inter alia that empirical distribution of the points (k,m)p−m with 0 ≤ k ≤ n < pm and (nk)≡a (mod⁡ p)s $left( {matrix{n cr k cr } } right) equiv aleft( {bmod ;p} right)^s $ (for (a, p) = 1) for m→∞ tends to the Hausdorff measure on the “p-adic Sierpiński gasket”, a fractals studied earlier by von Haeseler, Peitgen, and Skordev.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"61 1","pages":"151 - 161"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74161772","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 a Golay-Shapiro-Like Sequence 在Golay-Shapiro-Like Sequence上
Uniform distribution theory Pub Date : 2016-12-01 DOI: 10.1515/udt-2016-0021
J. Allouche
{"title":"On a Golay-Shapiro-Like Sequence","authors":"J. Allouche","doi":"10.1515/udt-2016-0021","DOIUrl":"https://doi.org/10.1515/udt-2016-0021","url":null,"abstract":"Abstract A recent paper by P. Lafrance, N. Rampersad, and R. Yee studies the sequence of occurrences of 10 as a scattered subsequence in the binary expansion of integers. They prove in particular that the summatory function of this sequence has the “root N” property, analogously to the summatory function of the Golay-Shapiro sequence. We prove here that the root N property does not hold if we twist the sequence by powers of a complex number of modulus one, hence showing a fundamental difference with the Golay-Shapiro sequence.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"15 1","pages":"205 - 210"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87350538","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
Yet Another Footnote to the Least Non Zero Digit of n! in Base 12 n最小非零位数的又一个注脚!12进制
Uniform distribution theory Pub Date : 2016-12-01 DOI: 10.1515/udt-2016-0018
J. Deshouillers
{"title":"Yet Another Footnote to the Least Non Zero Digit of n! in Base 12","authors":"J. Deshouillers","doi":"10.1515/udt-2016-0018","DOIUrl":"https://doi.org/10.1515/udt-2016-0018","url":null,"abstract":"Abstract We continue the study, initiated with Imre Ruzsa, of the last non zero digit ℓ12(n!) of n! in base 12, showing that for any a ∈ {3, 4, 6, 8, 9}, the set of those integers n for which ℓ12(n!) = a is not 3-automatic.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"32 1","pages":"163 - 167"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86023220","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
Convergence Results for r-Iterated Means of the Denominators of the Lüroth Series l<s:1>罗斯级数分母的r-迭代均值的收敛性结果
Uniform distribution theory Pub Date : 2016-12-01 DOI: 10.1515/udt-2016-0020
R. Giuliano
{"title":"Convergence Results for r-Iterated Means of the Denominators of the Lüroth Series","authors":"R. Giuliano","doi":"10.1515/udt-2016-0020","DOIUrl":"https://doi.org/10.1515/udt-2016-0020","url":null,"abstract":"Abstract In the present paper we extend two classic asymptotic results concerning convergence in probability and convergence in distribution for the denominators of the Lüroth series and obtain new theorems concerning the same two kinds of convergence for the r-iterated arithmetic means of such denominators. These results are extended to r-iterated weighted means.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"47 1","pages":"179 - 203"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88711876","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
Additive Energy and Irregularities of Distribution 加性能量与分布的不规则性
Uniform distribution theory Pub Date : 2016-08-24 DOI: 10.1515/udt-2017-0006
C. Aistleitner, G. Larcher
{"title":"Additive Energy and Irregularities of Distribution","authors":"C. Aistleitner, G. Larcher","doi":"10.1515/udt-2017-0006","DOIUrl":"https://doi.org/10.1515/udt-2017-0006","url":null,"abstract":"Abstract We consider strictly increasing sequences (an)n≥1 of integers and sequences of fractional parts ({anα})n≥1 where α ∈ R. We show that a small additive energy of (an)n≥1 implies that for almost all α the sequence ({anα})n≥1 has large discrepancy. We prove a general result, provide various examples, and show that the converse assertion is not necessarily true.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"1 1","pages":"107 - 99"},"PeriodicalIF":0.0,"publicationDate":"2016-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88565994","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
Uncanny Subsequence Selections That Generate Normal Numbers 不可思议的子序列选择产生正常的数字
Uniform distribution theory Pub Date : 2016-07-12 DOI: 10.1515/udt-2017-0015
J. Vandehey
{"title":"Uncanny Subsequence Selections That Generate Normal Numbers","authors":"J. Vandehey","doi":"10.1515/udt-2017-0015","DOIUrl":"https://doi.org/10.1515/udt-2017-0015","url":null,"abstract":"Abstract Given a real number 0.a1a2a3 . . . that is normal to base b, we examine increasing sequences ni so that the number 0.an1an2an3 . . . are normal to base b. Classically, it is known that if the ni form an arithmetic progression, then this will work. We give several more constructions including ni that are recursively defined based on the digits ai. 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)","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"95 1","pages":"65 - 75"},"PeriodicalIF":0.0,"publicationDate":"2016-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75852236","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
Uniform Distribution of the Sequence of Balancing Numbers Modulo m 以m为模的平衡数序列的均匀分布
Uniform distribution theory Pub Date : 2016-06-01 DOI: 10.1515/udt-2016-0002
P. Ray, Bijan Kumar Patel
{"title":"Uniform Distribution of the Sequence of Balancing Numbers Modulo m","authors":"P. Ray, Bijan Kumar Patel","doi":"10.1515/udt-2016-0002","DOIUrl":"https://doi.org/10.1515/udt-2016-0002","url":null,"abstract":"Abstract The balancing numbers and the balancers were introduced by Behera et al. in the year 1999, which were obtained from a simple diophantine equation. The goal of this paper is to investigate the moduli for which all the residues appear with equal frequency with a single period in the sequence of balancing numbers. Also, it is claimed that, the balancing numbers are uniformly distributed modulo 2, and this holds for all other powers of 2 as well. Further, it is shown that the balancing numbers are not uniformly distributed over odd primes.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"44 1","pages":"15 - 21"},"PeriodicalIF":0.0,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91372560","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
On Hausdorff Dimensions Related to Sets with Given Asymptotic and Gap Densities 关于具有给定渐近密度和间隙密度集的Hausdorff维
Uniform distribution theory Pub Date : 2016-06-01 DOI: 10.1515/udt-2016-0007
Ladislav Misík, J. Šustek, B. Volkmann
{"title":"On Hausdorff Dimensions Related to Sets with Given Asymptotic and Gap Densities","authors":"Ladislav Misík, J. Šustek, B. Volkmann","doi":"10.1515/udt-2016-0007","DOIUrl":"https://doi.org/10.1515/udt-2016-0007","url":null,"abstract":"Abstract For a set A of positive integers a1 < a2 < · · ·, let d(A), d¯(A) $overline d (A)$ denote its lower and upper asymptotic densities. The gap density is defined as λ(A)=lim⁡ supn→∞an+1an $lambda (A) = lim ;{rm sup} _{n to infty } {{a_{n + 1} } over {a_n }}$ . The paper investigates the class 𝒢(α, β, γ) of all sets A with d(A) = α, d¯(A)=β $overline d (A) = beta $ and λ(A) = γ for given α, β, γ with 0 ≤ α ≤ β ≤ 1 ≤ γ and αγ ≤ β. Using the classical dyadic mapping ϱ(A)=∑n=1∞χA(n)2n $varrho (A) = sumnolimits_{n = 1}^infty {{{chi _A (n)} over {2^n }}} $ , where χA is the characteristic function of A, the main result of the paper states that the ϱ-image set ϱ𝒢(α, β, γ) has the Hausdorff dimension dimϱ𝒢(α,β,γ)=min⁡{δ(α),δ(β),1γmax⁡σ∈[αγ,β]δ(σ)}, $$dim varrho cal {G}(alpha ,beta ,gamma ) = min left{ {delta (alpha ),delta (beta ), { 1 over gamma }mathop {max }limits_{sigma in [alpha gamma ,beta ]} delta (sigma )} right},$$ where δ is the entropy function δ(x)=−x log2 x−(1−x) log2 (1−x). $$delta (x) = - xlog _2 x - (1 - x);log _2 (1 - x).$$","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"19 1","pages":"141 - 157"},"PeriodicalIF":0.0,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73784663","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
Distribution of Leading Digits of Numbers 数字前导位数的分布
Uniform distribution theory Pub Date : 2016-06-01 DOI: 10.1515/udt-2016-0003
Y. Ohkubo, O. Strauch
{"title":"Distribution of Leading Digits of Numbers","authors":"Y. Ohkubo, O. Strauch","doi":"10.1515/udt-2016-0003","DOIUrl":"https://doi.org/10.1515/udt-2016-0003","url":null,"abstract":"Abstract Applying the theory of distribution functions of sequences we find the relative densities of the first digits also for sequences xn not satisfying Benford’s law. Especially for sequence xn = nr, n = 1, 2, . . . and xn=pnr $x_n = p_n^r $ , n = 1, 2, . . ., where pn is the increasing sequence of all primes and r > 0 is an arbitrary real. We also add rate of convergence to such densities.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"50 1","pages":"23 - 45"},"PeriodicalIF":0.0,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76329517","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
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