{"title":"Random Polynomials in Legendre Symbol Sequences","authors":"Katalin Gyarmati, Károly Müllner","doi":"10.2478/udt-2023-0006","DOIUrl":"https://doi.org/10.2478/udt-2023-0006","url":null,"abstract":"Abstract It is important in cryptographic applications that the “key” used should be generated from a random seed. Thus, if the Legendre symbol sequence generated by a polynomial (as proposed by Hoffstein and Lieman) is used, that is { (f(1)p),(f(2)p),(f(3)p),⋯,(f(p)p) }, left{ {left( {{{fleft( 1 right)} over p}} right),left( {{{fleft( 2 right)} over p}} right),left( {{{fleft( 3 right)} over p}} right), cdots ,left( {{{fleft( p right)} over p}} right)} right}, then it is important to choose the polynomial f “almost” at random. Goubin, Mauduit, and Sárközy presented some not very restrictive conditions on the polynomial f, but these conditions may not be satisfied if we choose a “truly” random polynomial. However, how can it be guaranteed that the pseudorandom measures of the sequence should be small for almost \"random\" polynomials? These semirandom polynomials will be constructed with as few modifications as necessary from a truly random polynomial.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"19 1","pages":"83 - 96"},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74849401","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}
{"title":"On the Expected ℒ2–Discrepancy of Jittered Sampling","authors":"Nathan Kirk, Florian Pausinger","doi":"10.2478/udt-2023-0005","DOIUrl":"https://doi.org/10.2478/udt-2023-0005","url":null,"abstract":"Abstract For m, d ∈ ℕ, a jittered sample of N = md points can be constructed by partitioning [0, 1]d into md axis-aligned equivolume boxes and placing one point independently and uniformly at random inside each box. We utilise a formula for the expected ℒ2−discrepancy of stratified samples stemming from general equivolume partitions of [0, 1]d which recently appeared, to derive a closed form expression for the expected ℒ2−discrepancy of a jittered point set for any m, d ∈ ℕ. As a second main result we derive a similar formula for the expected Hickernell ℒ2−discrepancy of a jittered point set which also takes all projections of the point set to lower dimensional faces of the unit cube into account.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"16 1","pages":"65 - 82"},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78284141","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}
{"title":"On a Reduced Component-by-Component Digit-by-Digit Construction of Lattice Point Sets","authors":"P. Kritzer, O. Osisiogu","doi":"10.2478/udt-2023-0007","DOIUrl":"https://doi.org/10.2478/udt-2023-0007","url":null,"abstract":"Abstract In this paper, we study an efficient algorithm for constructing point sets underlying quasi-Monte Carlo integration rules for weighted Korobov classes. The algorithm presented is a reduced fast component-by-component digit-by-digit (CBC-DBD) algorithm, which is useful for situations where the weights in the function space show a sufficiently fast decay. The advantage of the algorithm presented here is that the computational effort can be independent of the dimension of the integration problem to be treated if suitable assumptions on the integrand are met. By considering a reduced digit-by-digit construction, we allow an integration algorithm to be less precise with respect to the number of bits in those components of the problem that are considered less important. The new reduced CBC-DBD algorithm is designed to work for the construction of lattice point sets, and the corresponding integration rules (so-called lattice rules) can be used to treat functions in different kinds of function spaces. We show that the integration rules constructed by our algorithm satisfy error bounds of almost optimal convergence order. Furthermore, we give details on an efficient implementation such that we obtain a considerable speed-up of a previously known CBC-DBD algorithm that has been studied in the paper Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness by Ebert, Kritzer, Nuyens, and Osisiogu, published in the Journal of Complexity in 2021. This improvement is illustrated by numerical results.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"2 1","pages":"97 - 140"},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82120516","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}
{"title":"Notes on the Distribution of Roots Modulo Primes of a Polynomial IV","authors":"Y. Kitaoka","doi":"10.2478/udt-2023-0002","DOIUrl":"https://doi.org/10.2478/udt-2023-0002","url":null,"abstract":"Abstract For polynomials f (x),g1(x),g2(x)over ℤ, we report several observations about the density of primes p for which f (x) is fully splitting at p and { g1(r)p }<{ g2(r)p } left{ {{{{g_1}left( r right)} over p}} right} < left{ {{{{g_2}left( r right)} over p}} right} for some root r of f (x) ≡ 0 mod p.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"33 1","pages":"9 - 30"},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89196041","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}
{"title":"Copulas","authors":"O. Strauch, V. Baláž","doi":"10.2478/udt-2023-0009","DOIUrl":"https://doi.org/10.2478/udt-2023-0009","url":null,"abstract":"Abstract Two-dimensional distribution function g(x, y) defined in [0, 1]2 is called copula, if g(x, 1) = x and g(1,y)= y for every x, y. Similarly, s-dimensional copula is a distribution function g(x1,x2,...,xs) such that every k-dimensional face function g(1,…,1,xi1,1,…,1,xi2,1,…,1,xik,1,…,1) gleft( {1, ldots ,1,{x_{{i_1}}},1, ldots ,1,{x_{{i_2}}},1, ldots ,1,{x_{{i_k}}},1, ldots ,1} right) is equal to xi1 xi2 ...xik for some but fixed k. In this paper we summarize and extend all known parts of copulas. In this paper we use the following abbreviations: {x} — fractional part of x; {x} — x mod 1; [x] — integer part of x; u.d. — uniform distribution; d.f. — distribution function; a.d.f. — asymptotic distribution function; u.d.p. — uniform distribution preserving; step d.f. — step distribution function; a.e. — almost everywhere; #X — cardinality of the set X.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"76 1","pages":"147 - 200"},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83840412","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}
{"title":"Equidistribution of Continuous Functions Along Monotone Compact Covers","authors":"M. S. Monfared, Yihan Zhu","doi":"10.2478/udt-2023-0004","DOIUrl":"https://doi.org/10.2478/udt-2023-0004","url":null,"abstract":"Abstract We give a necessary and sufficient condition for equidistribution of continuous functions along monotone compact covers on locally compact spaces. We show the existence of equidistributed mappings along Bohr nets arising from group actions. Using almost periodic means, we give an analogue of Weyl’s equidistribution criterion for continuous functions with values in arbitrary topological groups. We prove van der Corput’s inequality on the lattice ℕm for vectors in Hilbert spaces, and use this inequality to extend Hlawka’s equidistribution theorem to functions on the lattice ℕm (m ≥ 1) with values in arbitrary topological groups.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"13 1","pages":"39 - 64"},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81202431","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}
{"title":"Refinement of the Theorem of Vahlen","authors":"Dinesh Sharma Bhattarai","doi":"10.2478/udt-2023-0001","DOIUrl":"https://doi.org/10.2478/udt-2023-0001","url":null,"abstract":"Abstract In 1895, Vahlen proved a theorem concerning a simultaneous approximation of a real number by its two consecutive convergent. In this paper, we will provide a sharper bound for the theorem.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"56 1","pages":"1 - 8"},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81502932","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}
{"title":"A Note on Well Distributed Sequences","authors":"Nikolay Moshohevitin","doi":"10.2478/udt-2023-0008","DOIUrl":"https://doi.org/10.2478/udt-2023-0008","url":null,"abstract":"Abstract We prove an easy statement about inhomogeneous approximation for non-singular vectors in metric theory of Diophantine approximation.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"20 1","pages":"141 - 146"},"PeriodicalIF":0.0,"publicationDate":"2023-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81953646","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}
{"title":"Kummer Theory for Multiquadratic or Quartic Cyclic Number Fields","authors":"Flavio Perissinotto, Antonella Perucca","doi":"10.2478/udt-2022-0017","DOIUrl":"https://doi.org/10.2478/udt-2022-0017","url":null,"abstract":"Abstract Let K be a number field which is multiquadratic or quartic cyclic. We prove several results about the Kummer extensions of K, namely concerning the intersection between the Kummer extensions and the cyclotomic extensions of K. For G a finitely generated subgroup of K×, we consider the cyclotomic-Kummer extensions K(ζnt,Gn)/K(ζnt) Kleft( {{zeta _{nt}},root n of G } right)/Kleft( {{zeta _{nt}}} right) for all positive integers n and t, and we describe an explicit finite procedure to compute at once the degree of all these extensions.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"11 1","pages":"165 - 194"},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86635451","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}
{"title":"A Note on the Distributions of (log n)mod 1","authors":"A. Berger","doi":"10.2478/udt-2022-0013","DOIUrl":"https://doi.org/10.2478/udt-2022-0013","url":null,"abstract":"Abstract For sequences sufficiently close to (a log n), with an arbitrary real constant a, this note describes the precise asymptotics of the associated empirical distributions modulo one, with respect to the Kantorovich metric as well as a discrepancy-style metric. In particular, the note demonstrates how these asymptotics depend on a in a delicate, discontinuous way. The results strengthen and complement known facts in the literature.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"30 1","pages":"77 - 100"},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81767377","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}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
