A list of Salem numbers less than $1.3$ is available on M. Mossinghoff's website (cite{MossinghoffList}). This list is certified complete up to degree $44$ in cite{MossinghoffRhinWu2008}, and it includes only one Salem number of degree $46$. The objective of the present work is to advance the understanding of Salem numbers by extending the list cite{MossinghoffList} through the provision of a list of Salem numbers less than the plastic constant, denoted by $eta$, which is approximately equal to $1.324718$. The algorithmic approach used is based on Integer Linear Programming.
{"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}
We study the set of combinatorial lengths of asymmetric periodic trajectories on the regular pentagon, proving a density-one version of a conjecture of Davis-Lelievre.
我们研究了正五边形上非对称周期轨迹的组合长度集，证明了戴维斯-勒里夫尔猜想的密度一版本。
{"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}
Let $f,ginmathbb{Z}[u_1,u_2]$ be binary quadratic forms. We provide upper bounds for the number of rational points $(u,v)inmathbb{P}^1(mathbb{Q})timesmathbb{P}^1(mathbb{Q})$ such that the ternary 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 bounds exist.
{"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}
Let $mathbb{F}_q$ be the finite field of order $q$, and $mathcal{A}$ a non-empty proper subset of $mathbb{F}_q$. Let $mathbf{M}$ be a random $m times n$ matrix of rank $r$ over $mathbb{F}_q$ taken with uniform distribution. It was proved recently by Sanna that as $m,n to infty$ and $r,q,mathcal{A}$ are fixed, the number of entries of $mathbf{M}$ in $mathcal{A}$ approaches a normal distribution. The question was raised as to whether or not one can still obtain a central limit theorem of some sort when $r$ goes to infinity in a way controlled by $m$ and $n$. In this paper we answer this question affirmatively.
{"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}
We show that the continuous 'etale cohomology groups $H^n_{mathrm{cont}}(X,mathbf{Q}_l(n))$ of smooth varieties $X$ over a finite field $k$ are spanned as $mathbf{Q}_l$-vector spaces by the $n$-th Milnor $K$-sheaf locally for the Zariski topology, for all $nge 0$. Here $l$ is a prime invertible in $k$. This is the first general unconditional result towards the conjectures of arXiv:math/9801017 (math.AG) which put together the Tate and the Beilinson conjectures relative to algebraic cycles on smooth projective $k$-varieties.
{"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}
Let $kgeqslant 2$ be an integer and let $lambda$ be the Liouville function. Given $k$ non-negative distinct integers $h_1,ldots,h_k$, the Chowla conjecture claims that $sum_{nleqslant x}lambda(n+h_1)cdotslambda(n+h_k)=o(x)$. An unconditional answer to this conjecture is yet to be found, and in this paper, we take a conditional approach. More precisely, we establish a bound for the sums $sum_{nleqslant x}lambda(n+h_1)cdotslambda(n+h_k)$ under the existence of Landau-Siegel zeroes. Our work constitutes an improvement over the previous related results of Germ'{a}n and K'{a}tai, Chinis, and Tao and Ter"av"ainen.
{"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}
Ben Krause, Hamed Mousavi, Terence Tao, Joni Teräväinen
We show that on a $sigma$-finite measure preserving system $X = (X,nu, T)$, the non-conventional ergodic averages $$ mathbb{E}_{n in [N]} Lambda(n) f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f in L^{p_1}(X)$, $g in L^{p_2}(X)$, and $1/p_1 + 1/p_2 leq 1$, where $P$ is a polynomial with integer coefficients of degree at least $2$. This had previously been established with the von Mangoldt weight $Lambda$ replaced by the constant weight $1$ by the first and third authors with Mirek, and by the M"obius weight $mu$ by the fourth author. The proof is based on combining tools from both of these papers, together with several Gowers norm and polynomial averaging operator estimates on approximants to the von Mangoldt function of ''Cram'er'' and ''Heath-Brown'' type.
我们证明，在$sigma$无限度量保全系统$X = (X,nu, T)$上，非常规遍历平均数$$ mathbb{E}_{n in [N]}Lambda(n)f(T^n x) g(T^{P(n)} x)$$ 对于 $f in L^{p_1}(X)$, $g in L^{p_2}(X)$, and $1/p_1 + 1/p_2 leq 1$, 其中 $P$ 是阶数至少为 2$ 的整数系数的偶项式，几乎无处不收敛。在此之前，第一位和第三位作者与米雷克一起用常数权$1$代替了冯-曼戈尔德权$Lambda$，第四位作者用莫比乌斯权$mu$代替了冯-曼戈尔德权$Lambda$。证明的基础是结合这两篇论文中的工具，以及关于''Cram'er'' 和 ''Heath-Brown''类型的冯-曼戈尔德函数近似值的几个高斯规范和多项式平均算子估计。
{"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}
We determine necessary and sufficient conditions for unicritical polynomials to be dynamically irreducible over finite fields. This result extends the results of Boston-Jones and Hamblen-Jones-Madhu regarding the dynamical irreducibility of particular families of unicritical polynomials. We also investigate dynamical irreducibility conditions for cubic and shifted linearized polynomials.
{"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}
We show the zero-density estimate [ N(zeta_{mathcal{P}}; alpha, T) ll T^{frac{4(1-alpha)}{3-2alpha-theta}}(log T)^{9} ] for Beurling zeta functions $zeta_{mathcal{P}}$ attached to Beurling generalized number systems with integers distributed as $N_{mathcal{P}}(x) = Ax + O(x^{theta})$. We also show a similar zero-density estimate for a broader class of general Dirichlet series, consider improvements conditional on finer pointwise or $L^{2k}$-bounds of $zeta_{mathcal{P}}$, and discuss some optimality questions.
{"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}
In recent work, Koukoulopoulos, Maynard and Yang proved an almost sharp quantitative bound for the Duffin-Schaeffer conjecture, using the Koukoulopoulos-Maynard technique of GCD graphs. This coincided with a simplification of the previous best known argument by Hauke, Vazquez and Walker, which avoided the use of the GCD graph machinery. In the present paper, we extend this argument to the new elements of the proof of Koukoulopoulos-Maynard-Yang. Combined with the work of Hauke-Vazquez-Walker, this provides a new proof of the almost sharp bound for the Duffin-Schaeffer conjecture, which avoids the use of GCD graphs entirely.
{"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}
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