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Lower bounds on the clique-chromatic numbers of some distance graphs 若干距离图的团色数的下界
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-06-23 DOI: 10.2140/moscow.2021.10.141
M. Koshelev
{"title":"Lower bounds on the clique-chromatic numbers of some distance graphs","authors":"M. Koshelev","doi":"10.2140/moscow.2021.10.141","DOIUrl":"https://doi.org/10.2140/moscow.2021.10.141","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":"41765244","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
Joint universality and simple a-points of Lerchzeta-functions Lerchzeta函数的联合泛性与简单a点
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-06-23 DOI: 10.2140/moscow.2021.10.121
H. Nagoshi
{"title":"Joint universality and simple a-points of Lerch\u0000zeta-functions","authors":"H. Nagoshi","doi":"10.2140/moscow.2021.10.121","DOIUrl":"https://doi.org/10.2140/moscow.2021.10.121","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":"46797532","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 approximation exponents for subspacesof ℝn 关于子空间的近似指数ℝn
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-06-08 DOI: 10.2140/moscow.2022.11.21
Elio Joseph
{"title":"On the approximation exponents for subspaces\u0000of ℝn","authors":"Elio Joseph","doi":"10.2140/moscow.2022.11.21","DOIUrl":"https://doi.org/10.2140/moscow.2022.11.21","url":null,"abstract":"This paper follows the generalisation of the classical theory of Diophantine approximation to subspaces of $mathbb{R}^n$ established by W. M. Schmidt in 1967. Let $A$ and $B$ be two subspaces of $mathbb{R}^n$ of respective dimensions $d$ and $e$ with $d+eleqslant n$. The proximity between $A$ and $B$ is measured by $t=min(d,e)$ canonical angles $0leqslant theta_1leqslant cdotsleqslant theta_tleqslant pi/2$; we set $psi_j(A,B)=sintheta_j$. If $B$ is a rational subspace, his complexity is measured by its height $H(B)=mathrm{covol}(Bcapmathbb{Z}^n)$. We denote by $mu_n(Avert e)_j$ the exponent of approximation defined as the upper bound (possibly equal to $+infty$) of the set of $beta>0$ such that the inequality $psi_j(A,B)leqslant H(B)^{-beta}$ holds for infinitely many rational subspaces $B$ of dimension $e$. We are interested in the minimal value $mathring{mu}_n(dvert e)_j$ taken by $mu_n(Avert e)_j$ when $A$ ranges through the set of subspaces of dimension $d$ of $mathbb{R}^n$ such that for all rational subspaces $B$ of dimension $e$ one has $dim (Acap B)<j$. We show that $mathring{mu}_4(2vert 2)_1=3$, $mathring{mu}_5(3vert 2)_1le 6$ and $mathring{mu}_{2d}(dvert ell)_1leqslant 2d^2/(2d-ell)$. We also prove a lower bound in the general case, which implies that $mathring{mu}_n(dvert d)_dxrightarrow[nto+infty]{} 1/d$.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42587677","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
Combinatorics of Serre weights in thepotentially Barsotti–Tate setting 潜在Barsotti-Tate设定中Serre权重的组合
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-05-10 DOI: 10.2140/moscow.2023.12.1
X. Caruso, Agnès David, Ariane M'ezard
{"title":"Combinatorics of Serre weights in the\u0000potentially Barsotti–Tate setting","authors":"X. Caruso, Agnès David, Ariane M'ezard","doi":"10.2140/moscow.2023.12.1","DOIUrl":"https://doi.org/10.2140/moscow.2023.12.1","url":null,"abstract":"Let $F$ be a finite unramified extension of $mathbb Q_p$ and $barrho$ be an absolutely irreducible mod~$p$ $2$-dimensional representation of the absolute Galois group of $F$. Let $t$ be a tame inertial type of $F$. We conjecture that the deformation space parametrizing the potentially Barsotti--Tate liftings of $barrho$ having type $t$ depends only on the Kisin variety attached to the situation, enriched with its canonical embedding into $(mathbb P^1)^f$ and its shape stratification. We give evidences towards this conjecture by proving that the Kisin variety determines the cardinality of the set of common Serre weights $D(t,barrho) = D(t) cap D(barrho)$. Besides, we prove that this dependance is nondecreasing (the smaller is the Kisin variety, the smaller is the number of common Serre weights) and compatible with products (if the Kisin variety splits as a product, so does the number of weights).","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49238091","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 statistics of pairs of logarithms of integers 关于整数对数对的统计
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-05-06 DOI: 10.2140/moscow.2022.11.335
Jouni Parkkonen, F. Paulin
{"title":"On the statistics of pairs of logarithms of integers","authors":"Jouni Parkkonen, F. Paulin","doi":"10.2140/moscow.2022.11.335","DOIUrl":"https://doi.org/10.2140/moscow.2022.11.335","url":null,"abstract":"We study the correlations of pairs of logarithms of positive integers at various scalings, either with trivial weigths or with weights given by the Euler function, proving the existence of pair correlation functions. We prove that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics. We prove total loss of mass phenomena at superlinear scalings, and Poissonian behaviour at sublinear scalings. The case of Euler weights has applications to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp neighborhood to itself in the modular curve $operatorname{PSL}_2(mathbb Z)backslashmathbb H^2_{mathbb R}$.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46597585","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
Rational approximations to two irrational numbers 两个无理数的有理数近似
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-04-07 DOI: 10.2140/moscow.2022.11.1
N. Shulga
{"title":"Rational approximations to two irrational numbers","authors":"N. Shulga","doi":"10.2140/moscow.2022.11.1","DOIUrl":"https://doi.org/10.2140/moscow.2022.11.1","url":null,"abstract":"For real ξ we consider the irrationality measure function ψξ(t) = min16q6t,q∈Z ||qξ||. We prove that in the case α± β / ∈ Z there exist arbitrary large values of t with ∣ ∣ ∣ 1 ψα(t) − 1 ψβ(t) ∣ ∣ ∣> √ 5 ","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44592799","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
Almost everywhere balanced sequences ofcomplexity 2n + 1 复杂性2n+1的几乎处处平衡序列
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-02-19 DOI: 10.2140/moscow.2022.11.287
J. Cassaigne, S'ebastien Labb'e, J. Leroy
{"title":"Almost everywhere balanced sequences of\u0000complexity 2n + 1","authors":"J. Cassaigne, S'ebastien Labb'e, J. Leroy","doi":"10.2140/moscow.2022.11.287","DOIUrl":"https://doi.org/10.2140/moscow.2022.11.287","url":null,"abstract":". We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set { 1 , 2 } N of directive sequences. For a given set C of two substitutions, we show that there exists a C -adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most 2 n +1 and is 2 n +1 if and only if the letter frequencies are rationally independent if and only if the C -adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that µ -almost every C -adic sequence is balanced, where µ is any shift-invariant ergodic Borel probability measure on { 1 , 2 } N giving a positive measure to the cylinder [12121212]. We also prove that the second Lyapunov exponent of the matrix cocycle associated with the measure µ is negative.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41944074","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
Minimal group determinants for dicyclic groups 双环群的最小群行列式
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-02-08 DOI: 10.2140/moscow.2021.10.235
B. Paudel, Christopher G. Pinner
{"title":"Minimal group determinants for dicyclic groups","authors":"B. Paudel, Christopher G. Pinner","doi":"10.2140/moscow.2021.10.235","DOIUrl":"https://doi.org/10.2140/moscow.2021.10.235","url":null,"abstract":"We determine the minimal non-trivial integer group determinant for the dicyclic group of order 4n when n is odd. We also discuss the set of all integer group determinants for the dicyclic groups of order 4p.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68080671","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 sparse perfect powers 关于稀疏完全幂
Moscow Journal of Combinatorics and Number Theory Pub Date : 2021-01-25 DOI: 10.2140/moscow.2021.10.261
A. Moscariello
{"title":"On sparse perfect powers","authors":"A. Moscariello","doi":"10.2140/moscow.2021.10.261","DOIUrl":"https://doi.org/10.2140/moscow.2021.10.261","url":null,"abstract":"This work is devoted to proving that, given an integer x ≥ 2, there are infinitely many perfect powers, coprime with x, having exactly k ≥ 3 non-zero digits in their base x representation, except for the case x = 2, k = 4, for which a known finiteness result by Corvaja and Zannier holds. Introduction Let k and x be positive integers, with x ≥ 2. In this work, we will study perfect powers having exactly k non-zero digits in their representation in a given basis x. These perfect powers are exactly (up to dividing by a suitable factor) the set solutions of the Diophantine equation (1) y = c0 + k−1","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47824446","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 the size of A+λA for algebraic λ 代数λ的A+λA的大小
Moscow Journal of Combinatorics and Number Theory Pub Date : 2020-09-30 DOI: 10.2140/moscow.2023.12.117
D. Krachun, F. Petrov
{"title":"On the size of A+λA for algebraic λ","authors":"D. Krachun, F. Petrov","doi":"10.2140/moscow.2023.12.117","DOIUrl":"https://doi.org/10.2140/moscow.2023.12.117","url":null,"abstract":"For a finite set $Asubset mathbb{R}$ and real $lambda$, let $A+lambda A:={a+lambda b :, a,bin A}$. Combining a structural theorem of Freiman on sets with small doubling constants together with a discrete analogue of Prekopa--Leindler inequality we prove a lower bound $|A+sqrt{2} A|geq (1+sqrt{2})^2|A|-O({|A|}^{1-varepsilon})$ which is essentially tight. We also formulate a conjecture about the value of $liminf |A+lambda A|/|A|$ for an arbitrary algebraic $lambda$. Finally, we prove a tight lower bound on the Lebesgue measure of $K+mathcal{T} K$ for a given linear operator $mathcal{T}in operatorname{End}(mathbb{R}^d)$ and a compact set $Ksubset mathbb{R}^d$ with fixed measure. This continuous result supports the conjecture and yields an upper bound in it.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68080776","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
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