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Notes on the Distribution of Roots Modulo a Prime of a Polynomial II 多项式模a素数的根分布的注释II
Uniform distribution theory Pub Date : 2019-06-01 DOI: 10.2478/udt-2019-0006
Y. Kitaoka
{"title":"Notes on the Distribution of Roots Modulo a Prime of a Polynomial II","authors":"Y. Kitaoka","doi":"10.2478/udt-2019-0006","DOIUrl":"https://doi.org/10.2478/udt-2019-0006","url":null,"abstract":"Abstract Let f (x) be a monic polynomial with integer coefficients and 0 ≤ r1 ≤ ··· ≤ rn <p its roots modulo a prime p. We generalize a conjecture on the distribution of roots ri with additional congruence relations ri ≡ Ri mod L from the case that f has no non-trivial linear relation among roots to the case that f has a non-trivial linear relation.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"40 1","pages":"104 - 87"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80609217","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 Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, II (Constructive Bounds) 二值数列相对于等差数列分布的不规则性，II(构造界)
Uniform distribution theory Pub Date : 2018-12-01 DOI: 10.2478/udt-2018-0008
Cécile Dartyge, Katalin Gyarmati, A. Sárközy
{"title":"On Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, II (Constructive Bounds)","authors":"Cécile Dartyge, Katalin Gyarmati, A. Sárközy","doi":"10.2478/udt-2018-0008","DOIUrl":"https://doi.org/10.2478/udt-2018-0008","url":null,"abstract":"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.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"76 1","pages":"1 - 21"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86524169","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
Sets of Bounded Remainder for The Billiard on A Square 正方形台球的有界余数集
Uniform distribution theory Pub Date : 2018-12-01 DOI: 10.2478/udt-2018-0011
I. Aichinger, G. Larcher
{"title":"Sets of Bounded Remainder for The Billiard on A Square","authors":"I. Aichinger, G. Larcher","doi":"10.2478/udt-2018-0011","DOIUrl":"https://doi.org/10.2478/udt-2018-0011","url":null,"abstract":"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.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"17 1","pages":"71 - 82"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78484945","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
Log-Like Functions and Uniform Distribution Modulo One 类对数函数与模1的均匀分布
Uniform distribution theory Pub Date : 2018-12-01 DOI: 10.2478/udt-2018-0013
M. Rehberg
{"title":"Log-Like Functions and Uniform Distribution Modulo One","authors":"M. Rehberg","doi":"10.2478/udt-2018-0013","DOIUrl":"https://doi.org/10.2478/udt-2018-0013","url":null,"abstract":"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.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"102 1","pages":"101 - 93"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80504343","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 M. B. Levin’s Proofs for The Exact Lower Discrepancy Bounds of Special Sequences and Point Sets (A Survey) M. B. Levin关于特殊序列和点集的精确差下界的证明(综述)
Uniform distribution theory Pub Date : 2018-12-01 DOI: 10.2478/udt-2018-0014
Lisa Kaltenböck, Wolfgang Stockinger
{"title":"On M. B. Levin’s Proofs for The Exact Lower Discrepancy Bounds of Special Sequences and Point Sets (A Survey)","authors":"Lisa Kaltenböck, Wolfgang Stockinger","doi":"10.2478/udt-2018-0014","DOIUrl":"https://doi.org/10.2478/udt-2018-0014","url":null,"abstract":"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.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"27 1","pages":"103 - 130"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72594329","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
On the Intriguing Search for Good Permutations 关于对好排列的有趣探索
Uniform distribution theory Pub Date : 2018-06-14 DOI: 10.2478/udt-2019-0005
Florian Pausinger
{"title":"On the Intriguing Search for Good Permutations","authors":"Florian Pausinger","doi":"10.2478/udt-2019-0005","DOIUrl":"https://doi.org/10.2478/udt-2019-0005","url":null,"abstract":"Abstract The intriguing search for permutations that generate generalised van der Corput sequences with exceptionally small discrepancy forms an important part of the research work of Henri Faure. On the occasion of Henri’s 80th birthday we aim to survey (some of) his contributions over the last four decades which considerably improved our understanding of one-dimensional van der Corput sequences and inspired a lot of related work. We recall and compare the different approaches in the search for generalised van der Corput sequences with low discrepancy, i.e., using a single generating permutation versus using a sequence of permutations. Throughout, we collect, sharpen and extend open questions which all stem from the extensive work of Henri and his coworkers and which will hopefully inspire more work in the future.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"40 1","pages":"53 - 86"},"PeriodicalIF":0.0,"publicationDate":"2018-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84045655","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
Weak Universality Theorem on the Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function from a Beatty Sequence 用Riemann - ζ函数的移位逼近解析函数的弱普适定理
Uniform distribution theory Pub Date : 2018-06-01 DOI: 10.1515/udt-2018-0007
Athanasios Sourmelidis
{"title":"Weak Universality Theorem on the Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function from a Beatty Sequence","authors":"Athanasios Sourmelidis","doi":"10.1515/udt-2018-0007","DOIUrl":"https://doi.org/10.1515/udt-2018-0007","url":null,"abstract":"Abstract In this paper, we prove a discrete analogue of Voronin’s early finite-dimensional approximation result with respect to terms from a given Beatty sequence and make use of Taylor approximation in order to derive a weak universality statement.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"11 1","pages":"131 - 146"},"PeriodicalIF":0.0,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88700383","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
Construction of Uniformly Distributed Linear Recurring Sequences Modulo Powers of 2 2的模幂的均匀分布线性循环序列的构造
Uniform distribution theory Pub Date : 2018-06-01 DOI: 10.1515/udt-2018-0006
T. Herendi
{"title":"Construction of Uniformly Distributed Linear Recurring Sequences Modulo Powers of 2","authors":"T. Herendi","doi":"10.1515/udt-2018-0006","DOIUrl":"https://doi.org/10.1515/udt-2018-0006","url":null,"abstract":"Abstract The aim of the present paper is to provide the background to construct linear recurring sequences with uniform distribution modulo 2s. The theory is developed and an algorithm based on the achieved results is given. The constructed sequences may have arbitrary large period length depending only on the computational power of the used machines.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"50 1","pages":"109 - 129"},"PeriodicalIF":0.0,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81787220","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
Motzkin’s Maximal Density and Related Chromatic Numbers 莫兹金最大密度与相关色数
Uniform distribution theory Pub Date : 2018-06-01 DOI: 10.1515/udt-2018-0002
Anshika Srivastava, R. K. Pandey, O. Prakash
{"title":"Motzkin’s Maximal Density and Related Chromatic Numbers","authors":"Anshika Srivastava, R. K. Pandey, O. Prakash","doi":"10.1515/udt-2018-0002","DOIUrl":"https://doi.org/10.1515/udt-2018-0002","url":null,"abstract":"Abstract This paper concerns the problem of determining or estimating the maximal upper density of the sets of nonnegative integers S whose elements do not differ by an element of a given set M of positive integers. We find some exact values and some bounds for the maximal density when the elements of M are generalized Fibonacci numbers of odd order. The generalized Fibonacci sequence of order r is a generalization of the well known Fibonacci sequence, where instead of starting with two predetermined terms, we start with r predetermined terms and each term afterwards is the sum of r preceding terms. We also derive some new properties of the generalized Fibonacci sequence of order r. Furthermore, we discuss some related coloring parameters of distance graphs generated by the set M.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"1 1","pages":"27 - 45"},"PeriodicalIF":0.0,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74849854","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
Questions Around the Thue-Morse Sequence 关于Thue-Morse序列的问题
Uniform distribution theory Pub Date : 2018-06-01 DOI: 10.1515/udt-2018-0001
M. Queffélec
{"title":"Questions Around the Thue-Morse Sequence","authors":"M. Queffélec","doi":"10.1515/udt-2018-0001","DOIUrl":"https://doi.org/10.1515/udt-2018-0001","url":null,"abstract":"Abstract We intend to unroll the surprizing properties of the Thue-Morse sequence with a harmonic analysis point of view, and mention in passing some related open questions.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"93 1","pages":"1 - 25"},"PeriodicalIF":0.0,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79439164","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: 7
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