arXiv - MATH - Number Theory最新文献_第5页

The section conjecture for the toric fundamental group over $p$-adic fields p$-adic 场上环基本群的截面猜想

arXiv - MATH - Number Theory Pub Date : 2024-09-12 DOI: arxiv-2409.07923

Giulio Bresciani

引用次数: 0

On determinants involving $(frac{j+k}p)pm(frac{j-k}p)$ 关于涉及 $(frac{j+k}p)pm(frac{j-k}p)$ 的行列式

arXiv - MATH - Number Theory Pub Date : 2024-09-12 DOI: arxiv-2409.08213

Deyi Chen, Zhi-Wei Sun

引用次数: 0

An inverse theorem for the Gowers $U^3$-norm relative to quadratic level sets 相对于二次水平集的高尔 U^3$ 准则的逆定理

arXiv - MATH - Number Theory Pub Date : 2024-09-12 DOI: arxiv-2409.07962

Sean Prendiville

{"title":"An inverse theorem for the Gowers $U^3$-norm relative to quadratic level sets","authors":"Sean Prendiville","doi":"arxiv-2409.07962","DOIUrl":"https://doi.org/arxiv-2409.07962","url":null,"abstract":"We prove an effective version of the inverse theorem for the Gowers\u0000$U^3$-norm for functions supported on high-rank quadratic level sets in finite\u0000vector spaces. For configurations controlled by the $U^3$-norm (complexity-two\u0000configurations), this enables one to run a density increment argument with\u0000respect to quadratic level sets, which are analogues of Bohr sets in the\u0000context of quadratic Fourier analysis on finite vector spaces. We demonstrate\u0000such an argument by deriving an exponential bound on the Ramsey number of\u0000three-term progressions which are the same colour as their common difference\u0000(``Brauer quadruples''), a result we have been unable to establish by other\u0000means. Our methods also yield polylogarithmic bounds on the density of sets lacking\u0000translation-invariant configurations of complexity two. Such bounds for\u0000four-term progressions were obtained by Green and Tao using a simpler\u0000weak-regularity argument. In an appendix, we give an example of how to\u0000generalise Green and Tao's argument to other translation-invariant\u0000configurations of complexity two. However, this crucially relies on an estimate\u0000coming from the Croot-Lev-Pach polynomial method, which may not be applicable\u0000to all systems of complexity two. Hence running a density increment with\u0000respect to quadratic level sets may still prove useful for such problems. It\u0000may also serve as a model for running density increments on more general\u0000nil-Bohr sets, with a view to effectivising other Szemer'edi-type theorems.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203820","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 Algebraic Proof of Hrushovski's Theorem 赫鲁晓夫斯基定理的代数证明

arXiv - MATH - Number Theory Pub Date : 2024-09-12 DOI: arxiv-2409.08370

Thomas Wisson

引用次数: 0

General Dynamics and Generation Mapping for Collatz-type Sequences 通用动力和科拉茨型序列的世代映射

arXiv - MATH - Number Theory Pub Date : 2024-09-12 DOI: arxiv-2409.07929

Gaurav Goyal

{"title":"General Dynamics and Generation Mapping for Collatz-type Sequences","authors":"Gaurav Goyal","doi":"arxiv-2409.07929","DOIUrl":"https://doi.org/arxiv-2409.07929","url":null,"abstract":"Let an odd integer (mathcal{X}) be expressed as $left{sumlimits_{M >\u0000m}b_M2^Mright}+2^m-1,$ where $b_Min{0,1}$ and $2^m-1$ is referred to as\u0000the Governor. In Collatz-type functions, a high index Governor is eventually\u0000reduced to $2^1-1$. For the $3mathcal{Z}+1$ sequence, the Governor occurring\u0000in the Trivial cycle is $2^1-1$, while for the $5mathcal{Z}+1$ sequence, the\u0000Trivial Governors are $2^2-1$ and $2^1-1$. Therefore, in these specific\u0000sequences, the Collatz function reduces the Governor $2^m - 1$ to the Trivial\u0000Governor $2^{mathcal{T}} - 1$. Once this Trivial Governor is reached, it can\u0000evolve to a higher index Governor through interactions with other terms. This\u0000feature allows $mathcal{X}$ to reappear in a Collatz-type sequence, since $2^m\u0000- 1 = 2^{m - 1} + cdots + 2^{mathcal{T} + 1} +\u00002^{mathcal{T}}+(2^{mathcal{T}}-1).$ Thus, if $mathcal{X}$ reappears, at\u0000least one odd ancestor of $left{sumlimits_{M >\u0000m}b_M2^Mright}+2^{m-1}+cdots+2^{mathcal{T}+1}+2^{mathcal{T}}+(2^{mathcal{T}}-1)$\u0000must have the Governor $2^m-1$. Ancestor mapping shows that all odd ancestors\u0000of $mathcal{X}$ have the Trivial Governor for the respective Collatz sequence.\u0000This implies that odd integers that repeat in the $3mathcal{Z} + 1$ sequence\u0000have the Governor $2^1 - 1$, while those forming a repeating cycle in the\u0000$5mathcal{Z} + 1$ sequence have either $2^2 - 1$ or $2^1 - 1$ as the Governor.\u0000Successor mapping for the $3mathcal{Z} + 1$ sequence further indicates that\u0000there are no auxiliary cycles, as the Trivial Governor is always transformed\u0000into a different index Governor. Similarly, successor mapping for the\u0000$5mathcal{Z} + 1$ sequence reveals that the smallest odd integers forming an\u0000auxiliary cycle are smaller than $2^5$. Finally, attempts to identify integers\u0000that diverge for the $3mathcal{Z} + 1$ sequence suggest that no such integers\u0000exist.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203798","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 note on local formulae for the parity of Selmer ranks 关于塞尔默等级奇偶性局部公式的说明

arXiv - MATH - Number Theory Pub Date : 2024-09-12 DOI: arxiv-2409.08034

Adam Morgan

引用次数: 0

Lower order terms in the shape of cubic fields 立方场形状中的低阶项

arXiv - MATH - Number Theory Pub Date : 2024-09-12 DOI: arxiv-2409.08417

Robert Hough, Eun Hye Lee

引用次数: 0

No proper generalized quadratic forms are universal over quadratic fields 没有适当的广义二次型是在二次域上通用的

arXiv - MATH - Number Theory Pub Date : 2024-09-12 DOI: arxiv-2409.07941

Ondřej ChwiedziukCharles University, Matěj DoležálekCharles University, Simona HlavinkováCharles University, Emma PěchoučkováCharles University, Zdeněk PezlarCharles University, Om PrakashCharles University, Anna RůžičkováCharles University, Mikuláš ZindulkaCharles University

引用次数: 0

A probabilistic proof of a recurrence relation for sums of values of degenerate falling factorials 退化下降阶乘值之和递推关系的概率证明

arXiv - MATH - Number Theory Pub Date : 2024-09-12 DOI: arxiv-2409.07742

Taekyun Kim, Dae san Kim

引用次数: 0

Intersection of orbits of loxodromic automorphisms of affine surfaces 仿射曲面的loxodromic自动形的轨道交集

arXiv - MATH - Number Theory Pub Date : 2024-09-12 DOI: arxiv-2409.07826

Marc Abboud

引用次数: 0