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Salem numbers less than the plastic constant 小于塑性常数的塞勒姆数
arXiv - MATH - Number Theory Pub Date : 2024-09-17 DOI: arxiv-2409.11159
Jean-Marc Sac-Épée
{"title":"Salem numbers less than the plastic constant","authors":"Jean-Marc Sac-Épée","doi":"arxiv-2409.11159","DOIUrl":"https://doi.org/arxiv-2409.11159","url":null,"abstract":"A list of Salem numbers less than $1.3$ is available on M. Mossinghoff's\u0000website (cite{MossinghoffList}). This list is certified complete up to degree\u0000$44$ in cite{MossinghoffRhinWu2008}, and it includes only one Salem number of\u0000degree $46$. The objective of the present work is to advance the understanding\u0000of Salem numbers by extending the list cite{MossinghoffList} through the\u0000provision of a list of Salem numbers less than the plastic constant, denoted by\u0000$eta$, which is approximately equal to $1.324718$. The algorithmic approach\u0000used is based on Integer Linear Programming.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259848","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 Local-Global Conjecture for Combinatorial Period Lengths of Closed Billiards on the Regular Pentagon 关于正五边形上封闭台球组合周期长度的局部-全局猜想
arXiv - MATH - Number Theory Pub Date : 2024-09-16 DOI: arxiv-2409.10682
Alex Kontorovich, Xin Zhang
{"title":"On the Local-Global Conjecture for Combinatorial Period Lengths of Closed Billiards on the Regular Pentagon","authors":"Alex Kontorovich, Xin Zhang","doi":"arxiv-2409.10682","DOIUrl":"https://doi.org/arxiv-2409.10682","url":null,"abstract":"We study the set of combinatorial lengths of asymmetric periodic trajectories\u0000on the regular pentagon, proving a density-one version of a conjecture of\u0000Davis-Lelievre.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259906","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
Local solubility of a family of ternary conics over a biprojective base I 双射基 I 上三元圆锥族的局部可溶性
arXiv - MATH - Number Theory Pub Date : 2024-09-16 DOI: arxiv-2409.10688
Cameron Wilson
{"title":"Local solubility of a family of ternary conics over a biprojective base I","authors":"Cameron Wilson","doi":"arxiv-2409.10688","DOIUrl":"https://doi.org/arxiv-2409.10688","url":null,"abstract":"Let $f,ginmathbb{Z}[u_1,u_2]$ be binary quadratic forms. We provide upper\u0000bounds for the number of rational points\u0000$(u,v)inmathbb{P}^1(mathbb{Q})timesmathbb{P}^1(mathbb{Q})$ such that the\u0000ternary conic [ X_{(u,v)}: f(u_1,u_2)x^2 + g(v_1,v_2)y^2 = z^2 ] has a rational point. We also give some conditions under which lower\u0000bounds exist.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269315","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
The central limit theorem for entries of random matrices with specific rank over finite fields 有限域上特定秩随机矩阵条目的中心极限定理
arXiv - MATH - Number Theory Pub Date : 2024-09-16 DOI: arxiv-2409.10412
Chin Hei Chan, Maosheng Xiong
{"title":"The central limit theorem for entries of random matrices with specific rank over finite fields","authors":"Chin Hei Chan, Maosheng Xiong","doi":"arxiv-2409.10412","DOIUrl":"https://doi.org/arxiv-2409.10412","url":null,"abstract":"Let $mathbb{F}_q$ be the finite field of order $q$, and $mathcal{A}$ a\u0000non-empty proper subset of $mathbb{F}_q$. Let $mathbf{M}$ be a random $m\u0000times n$ matrix of rank $r$ over $mathbb{F}_q$ taken with uniform\u0000distribution. It was proved recently by Sanna that as $m,n to infty$ and\u0000$r,q,mathcal{A}$ are fixed, the number of entries of $mathbf{M}$ in\u0000$mathcal{A}$ approaches a normal distribution. The question was raised as to\u0000whether or not one can still obtain a central limit theorem of some sort when\u0000$r$ goes to infinity in a way controlled by $m$ and $n$. In this paper we\u0000answer this question affirmatively.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259909","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
An $l$-adic norm residue epimorphism theorem l$-adic常模残差上变定理
arXiv - MATH - Number Theory Pub Date : 2024-09-16 DOI: arxiv-2409.10248
Bruno Kahn
{"title":"An $l$-adic norm residue epimorphism theorem","authors":"Bruno Kahn","doi":"arxiv-2409.10248","DOIUrl":"https://doi.org/arxiv-2409.10248","url":null,"abstract":"We show that the continuous 'etale cohomology groups\u0000$H^n_{mathrm{cont}}(X,mathbf{Q}_l(n))$ of smooth varieties $X$ over a finite\u0000field $k$ are spanned as $mathbf{Q}_l$-vector spaces by the $n$-th Milnor\u0000$K$-sheaf locally for the Zariski topology, for all $nge 0$. Here $l$ is a\u0000prime invertible in $k$. This is the first general unconditional result towards\u0000the conjectures of arXiv:math/9801017 (math.AG) which put together the Tate and\u0000the Beilinson conjectures relative to algebraic cycles on smooth projective\u0000$k$-varieties.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259915","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
The Chowla conjecture and Landau-Siegel zeroes 乔拉猜想和兰道-西格尔零点
arXiv - MATH - Number Theory Pub Date : 2024-09-16 DOI: arxiv-2409.10663
Mikko Jaskari, Stelios Sachpazis
{"title":"The Chowla conjecture and Landau-Siegel zeroes","authors":"Mikko Jaskari, Stelios Sachpazis","doi":"arxiv-2409.10663","DOIUrl":"https://doi.org/arxiv-2409.10663","url":null,"abstract":"Let $kgeqslant 2$ be an integer and let $lambda$ be the Liouville function.\u0000Given $k$ non-negative distinct integers $h_1,ldots,h_k$, the Chowla\u0000conjecture claims that $sum_{nleqslant\u0000x}lambda(n+h_1)cdotslambda(n+h_k)=o(x)$. An unconditional answer to this\u0000conjecture is yet to be found, and in this paper, we take a conditional\u0000approach. More precisely, we establish a bound for the sums $sum_{nleqslant\u0000x}lambda(n+h_1)cdotslambda(n+h_k)$ under the existence of Landau-Siegel\u0000zeroes. Our work constitutes an improvement over the previous related results\u0000of Germ'{a}n and K'{a}tai, Chinis, and Tao and Ter\"av\"ainen.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259907","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
Pointwise convergence of bilinear polynomial averages over the primes 素数上双线性多项式平均数的点收敛性
arXiv - MATH - Number Theory Pub Date : 2024-09-16 DOI: arxiv-2409.10510
Ben Krause, Hamed Mousavi, Terence Tao, Joni Teräväinen
{"title":"Pointwise convergence of bilinear polynomial averages over the primes","authors":"Ben Krause, Hamed Mousavi, Terence Tao, Joni Teräväinen","doi":"arxiv-2409.10510","DOIUrl":"https://doi.org/arxiv-2409.10510","url":null,"abstract":"We show that on a $sigma$-finite measure preserving system $X = (X,nu, T)$,\u0000the non-conventional ergodic averages $$ mathbb{E}_{n in [N]} Lambda(n)\u0000f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f in\u0000L^{p_1}(X)$, $g in L^{p_2}(X)$, and $1/p_1 + 1/p_2 leq 1$, where $P$ is a\u0000polynomial with integer coefficients of degree at least $2$. This had\u0000previously been established with the von Mangoldt weight $Lambda$ replaced by\u0000the constant weight $1$ by the first and third authors with Mirek, and by the\u0000M\"obius weight $mu$ by the fourth author. The proof is based on combining\u0000tools from both of these papers, together with several Gowers norm and\u0000polynomial averaging operator estimates on approximants to the von Mangoldt\u0000function of ''Cram'er'' and ''Heath-Brown'' type.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259914","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
Dynamical Irreducibility of Certain Families of Polynomials over Finite Fields 有限域上多项式某些族的动态不可约性
arXiv - MATH - Number Theory Pub Date : 2024-09-16 DOI: arxiv-2409.10467
Tori Day, Rebecca DeLand, Jamie Juul, Cigole Thomas, Bianca Thompson, Bella Tobin
{"title":"Dynamical Irreducibility of Certain Families of Polynomials over Finite Fields","authors":"Tori Day, Rebecca DeLand, Jamie Juul, Cigole Thomas, Bianca Thompson, Bella Tobin","doi":"arxiv-2409.10467","DOIUrl":"https://doi.org/arxiv-2409.10467","url":null,"abstract":"We determine necessary and sufficient conditions for unicritical polynomials\u0000to be dynamically irreducible over finite fields. This result extends the\u0000results of Boston-Jones and Hamblen-Jones-Madhu regarding the dynamical\u0000irreducibility of particular families of unicritical polynomials. We also\u0000investigate dynamical irreducibility conditions for cubic and shifted\u0000linearized polynomials.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269311","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 zero-density estimates for Beurling zeta functions 关于 Beurling 兹塔函数的零密度估计
arXiv - MATH - Number Theory Pub Date : 2024-09-16 DOI: arxiv-2409.10051
Frederik Broucke
{"title":"On zero-density estimates for Beurling zeta functions","authors":"Frederik Broucke","doi":"arxiv-2409.10051","DOIUrl":"https://doi.org/arxiv-2409.10051","url":null,"abstract":"We show the zero-density estimate [ N(zeta_{mathcal{P}}; alpha, T) ll\u0000T^{frac{4(1-alpha)}{3-2alpha-theta}}(log T)^{9} ] for Beurling zeta\u0000functions $zeta_{mathcal{P}}$ attached to Beurling generalized number systems\u0000with integers distributed as $N_{mathcal{P}}(x) = Ax + O(x^{theta})$. We also\u0000show a similar zero-density estimate for a broader class of general Dirichlet\u0000series, consider improvements conditional on finer pointwise or $L^{2k}$-bounds\u0000of $zeta_{mathcal{P}}$, and discuss some optimality questions.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259911","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
Almost-Sharp Quantitative Duffin-Shaeffer without GCD Graphs 无 GCD 图形的近乎锐利的定量达芬-谢弗
arXiv - MATH - Number Theory Pub Date : 2024-09-16 DOI: arxiv-2409.10386
Santiago Vazquez
{"title":"Almost-Sharp Quantitative Duffin-Shaeffer without GCD Graphs","authors":"Santiago Vazquez","doi":"arxiv-2409.10386","DOIUrl":"https://doi.org/arxiv-2409.10386","url":null,"abstract":"In recent work, Koukoulopoulos, Maynard and Yang proved an almost sharp\u0000quantitative bound for the Duffin-Schaeffer conjecture, using the\u0000Koukoulopoulos-Maynard technique of GCD graphs. This coincided with a\u0000simplification of the previous best known argument by Hauke, Vazquez and\u0000Walker, which avoided the use of the GCD graph machinery. In the present paper,\u0000we extend this argument to the new elements of the proof of\u0000Koukoulopoulos-Maynard-Yang. Combined with the work of Hauke-Vazquez-Walker,\u0000this provides a new proof of the almost sharp bound for the Duffin-Schaeffer\u0000conjecture, which avoids the use of GCD graphs entirely.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269313","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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