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Moscow Journal of Combinatorics and Number Theory最新文献

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Diophantine sets and Dirichlet improvability 丢番图集和狄利克雷可改进性
Moscow Journal of Combinatorics and Number Theory Pub Date : 2022-01-26 DOI: 10.2140/moscow.2022.11.189
Antoine Marnat
{"title":"Diophantine sets and Dirichlet improvability","authors":"Antoine Marnat","doi":"10.2140/moscow.2022.11.189","DOIUrl":"https://doi.org/10.2140/moscow.2022.11.189","url":null,"abstract":"This note pushes further the discussion about relations between Dirichlet improvable, badly approximable and singular points held in recent joint work with Beresnevich, Guan, Velani and Ramirez, by considering Diophantine sets extending the notion of badly approximability.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47914600","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
An improved convergence case for Diophantine approximations on IFS fractals IFS分形上Diophantine近似的改进收敛情形
Moscow Journal of Combinatorics and Number Theory Pub Date : 2022-01-23 DOI: 10.2140/moscow.2023.12.97
Itamar Cohen-Matalon
{"title":"An improved convergence case for Diophantine approximations on IFS fractals","authors":"Itamar Cohen-Matalon","doi":"10.2140/moscow.2023.12.97","DOIUrl":"https://doi.org/10.2140/moscow.2023.12.97","url":null,"abstract":"The objective of this paper is to (partially) address the issue of finding an analogue to Khintchine's theorem for IFS Fractals. We study the convergence case for Diophantine approximations, and show an improved result for higher dimensions. This matter has been previously studied by Pollington and Velani in arXiv:math/0401149. Pollington and Velani show a similar result to the one in this paper (a Khinchine convergence case) and we shall show how our result is an improvement in the higher dimensional cases.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48036432","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 generalization of a theorem of White 一个White定理的推广
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-12-31 DOI: 10.2140/moscow.2021.10.281
V. Batyrev, Johannes Hofscheier
{"title":"A generalization of a theorem of White","authors":"V. Batyrev, Johannes Hofscheier","doi":"10.2140/moscow.2021.10.281","DOIUrl":"https://doi.org/10.2140/moscow.2021.10.281","url":null,"abstract":"","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42144987","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
Visibility properties of spiral sets 螺旋集的可见性
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-11-02 DOI: 10.2140/moscow.2022.11.149
F. Adiceam, Ioannis Tsokanos
{"title":"Visibility properties of spiral sets","authors":"F. Adiceam, Ioannis Tsokanos","doi":"10.2140/moscow.2022.11.149","DOIUrl":"https://doi.org/10.2140/moscow.2022.11.149","url":null,"abstract":". A spiral in R d +1 is defined as a set of the form { d +1 √ n · u n } n ≥ 1 , where ( u n ) n ≥ 1 is a spherical sequence. Such point sets have been extensively studied, in particular in the planar case d = 1, as they then serve as natural models describing phyllotactic structures (i.e. structures representing configurations of leaves on a plant stem). Recent progress in this theory provides a fine analysis of the distribution of spirals (e.g., their covering and packing radii). Here, various concepts of visiblity from discrete geometry are employed to characterise density properties of such point sets. More pre-cisely, necessary an sufficient conditions are established for a spiral to be (1) an orchard (a “homogeneous” density property defined by P`olya), (2) a uniform orchard (a concept introduced in this work), (3) a set with no visible point (implying that the point set is dense enough in a suitable sense) and (4) a dense forest (a quantitative and uniform refinement of the previous concept).","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41415488","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
Improved constants for effective irrationality measures from hypergeometric functions 改进了超几何函数中有效无理性测度的常数
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-11-01 DOI: 10.2140/moscow.2022.11.161
P. Voutier
{"title":"Improved constants for effective irrationality measures from hypergeometric functions","authors":"P. Voutier","doi":"10.2140/moscow.2022.11.161","DOIUrl":"https://doi.org/10.2140/moscow.2022.11.161","url":null,"abstract":". In this paper, we simplify and improve the constant, c , that appears in effective irrationality measures, | ( a/b ) m/n − p/q | > c | q | − ( κ +1) , obtained from the hypergeometric method for a/b near 1. The dependence of c on both | a | in our result is best possible (as is the dependence on n in many cases). For some applications, the dependence of this constant on | a | becomes important. We also establish some new inequalities for hypergeometric functions that are useful in other diophantine settings.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44617890","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 representation of integers by binaryforms defined by means of the relation (x + yi)n= Rn(x,y) + Jn(x,y)i 关于由关系式（x+yi）n=Rn（x，y）+Jn（x、y）i定义的二进制形式表示整数
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-10-11 DOI: 10.2140/moscow.2022.11.71
A. Mosunov
{"title":"On the representation of integers by binary\u0000forms defined by means of the relation (x + yi)n= Rn(x,y) + Jn(x,y)i","authors":"A. Mosunov","doi":"10.2140/moscow.2022.11.71","DOIUrl":"https://doi.org/10.2140/moscow.2022.11.71","url":null,"abstract":"Let F be a binary form with integer coefficients, degree d ≥ 3 and nonzero discriminant. Let RF (Z) denote the number of integers of absolute value at most Z which are represented by F . In 2019 Stewart and Xiao proved that RF (Z) ∼ CFZ 2/d for some positive number CF . We compute CRn and CJn for the binary forms Rn(x, y) and Jn(x, y) defined by means of the relation (x+ yi) = Rn(x, y) + Jn(x, y)i, where the variables x and y are real.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42610340","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
Cyclic and well-rounded lattices 循环格和全面格
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-10-10 DOI: 10.2140/moscow.2022.11.79
L. Fukshansky, David Kogan
{"title":"Cyclic and well-rounded lattices","authors":"L. Fukshansky, David Kogan","doi":"10.2140/moscow.2022.11.79","DOIUrl":"https://doi.org/10.2140/moscow.2022.11.79","url":null,"abstract":"We focus on two important classes of lattices, the well-rounded and the cyclic. We show that every well-rounded lattice in the plane is similar to a cyclic lattice, and use this cyclic parameterization to count planar wellrounded similarity classes defined over a fixed number field with respect to height. We then investigate cyclic properties of the irreducible root lattices in arbitrary dimensions, in particular classifying those that are simple cyclic, i.e. generated by rotation shifts of a single vector. Finally, we classify cyclic, simple cyclic and well-rounded cyclic lattices coming from rings of integers of Galois algebraic number fields.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48843272","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
Uniformly distributed sequences generated by agreedy minimization of the L2 discrepancy 由L2差的一致最小化生成的均匀分布序列
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-09-13 DOI: 10.2140/moscow.2022.11.215
Ralph Kritzinger
{"title":"Uniformly distributed sequences generated by a\u0000greedy minimization of the L2 discrepancy","authors":"Ralph Kritzinger","doi":"10.2140/moscow.2022.11.215","DOIUrl":"https://doi.org/10.2140/moscow.2022.11.215","url":null,"abstract":"The aim of this paper is to develop greedy algorithms which generate uniformly distributed sequences in the d -dimensional unit cube [0 , 1] d . The figures of merit are three different variants of L 2 discrepancy. Theoretical results along with numerical experiments suggest that the resulting sequences have excellent distribution properties. The approach we follow here is motivated by recent work of Steinerberger and Pausinger who consider similar greedy algorithms, where they minimize functionals that can be related to the star discrepancy or energy of point sets. In contrast to many greedy algorithms where the resulting elements of the sequence can only be given numerically, we will find that in the one-dimensional case our algorithms yield rational numbers which we can describe precisely. In particular, we will observe that any initial segment of a sequence in [0 , 1) can be naturally extended to a uniformly distributed sequence where all subsequent elements are of the form x N = 2 n − 1 2 N for some n ∈ { 1 , . . . , N } . We will also investigate the dependence of the L 2 discrepancy of the resulting sequences on the dimension d .","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45780273","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
Abundance of Dirichlet-improvable pairs with respect to arbitrary norms 关于任意范数的Dirichlet可改进对的丰富性
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-07-21 DOI: 10.2140/moscow.2022.11.97
D. Kleinbock, Anurag Rao
{"title":"Abundance of Dirichlet-improvable pairs with respect to arbitrary norms","authors":"D. Kleinbock, Anurag Rao","doi":"10.2140/moscow.2022.11.97","DOIUrl":"https://doi.org/10.2140/moscow.2022.11.97","url":null,"abstract":"In a recent paper of Akhunzhanov and Shatskov the two-dimensional Dirichlet spectrum with respect to Euclidean norm was defined. We consider an analogous definition for arbitrary norms on $mathbb{R}^2$ and prove that, for each such norm, the set of Dirichlet improvable pairs contains the set of badly approximable pairs, hence is hyperplane absolute winning. To prove this we make a careful study of some classical results in the geometry of numbers due to Chalk--Rogers and Mahler to establish a Haj'{o}s--Minkowski type result for the critical locus of a cylinder. As a corollary, using a recent result of the first named author with Mirzadeh, we conclude that for any norm on $mathbb{R}^2$ the top of the Dirichlet spectrum is not an isolated point.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46869671","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
Voronoi conjecture for special free parallelotopes 特殊自由平行四边形的Voronoi猜想
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-06-23 DOI: 10.2140/moscow.2021.10.83
V. Grishukhin
{"title":"Voronoi conjecture for special free parallelotopes","authors":"V. Grishukhin","doi":"10.2140/moscow.2021.10.83","DOIUrl":"https://doi.org/10.2140/moscow.2021.10.83","url":null,"abstract":"","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46804433","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
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