Mathematical Proceedings of the Cambridge Philosophical Society最新文献

{"title":"Some torsion-free solvable groups with few subquotients","authors":"Adrien Le Boudec, Nicolás Matte Bon","doi":"10.1017/s0305004123000506","DOIUrl":"https://doi.org/10.1017/s0305004123000506","url":null,"abstract":"Abstract We construct finitely generated torsion-free solvable groups G that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of G are virtually abelian. In particular all finitely generated metabelian subgroups of G are virtually abelian. The existence of such groups shows that there is no “torsion-free version” of P. Kropholler’s theorem, which characterises solvable groups of infinite rank via their metabelian subquotients.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135789980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multiple recurrence and popular differences for polynomial patterns in rings of integers","authors":"ETHAN ACKELSBERG, VITALY BERGELSON","doi":"10.1017/s030500412300049x","DOIUrl":"https://doi.org/10.1017/s030500412300049x","url":null,"abstract":"Abstract We demonstrate that the phenomenon of popular differences (aka the phenomenon of large intersections) holds for natural families of polynomial patterns in rings of integers of number fields. If K is a number field with ring of integers $mathcal{O}_K$ and $E subseteq mathcal{O}_K$ has positive upper Banach density $d^*(E) = delta > 0$ , we show, inter alia : (1) if $p(x) in K[x]$ is an intersective polynomial (i.e., p has a root modulo m for every $m in mathcal{O}_K$ ) with $p(mathcal{O}_K) subseteq mathcal{O}_K$ and $r, s in mathcal{O}_K$ are distinct and nonzero, then for any $varepsilon > 0$ , there is a syndetic set $S subseteq mathcal{O}_K$ such that for any $n in S$ , begin{align*} d^* left( left{ x in mathcal{O}_K ;:; {x, x + rp(n), x + sp(n)} subseteq E right} right) > delta^3 - varepsilon. end{align*} Moreover, if ${s}/{r} in mathbb{Q}$ , then there are syndetically many $n in mathcal{O}_K$ such that begin{align*} d^* left( left{ x in mathcal{O}_K ;:; {x, x + rp(n), x + sp(n), x + (r+s)p(n)} subseteq E right} right) > delta^4 - varepsilon; end{align*} (2) if ${p_1, dots, p_k} subseteq K[x]$ is a jointly intersective family (i.e., $p_1, dots, p_k$ have a common root modulo m for every $m in mathcal{O}_K$ ) of linearly independent polynomials with $p_i(mathcal{O}_K) subseteq mathcal{O}_K$ , then there are syndetically many $n in mathcal{O}_K$ such that begin{align*} d^* left( left{ x in mathcal{O}_K ;:; {x, x + p_1(n), dots, x + p_k(n)} subseteq E right} right) > delta^{k+1} - varepsilon. end{align*} These two results generalise and extend previous work of Frantzikinakis and Kra [ 21 ] and Franztikinakis [ 19 ] on polynomial configurations in $mathbb{Z}$ and build upon recent work of the authors and Best [ 2 ] on linear patterns in general abelian groups. The above combinatorial results follow from multiple recurrence results in ergodic theory via a version of Furstenberg’s correspondence principle. The ergodic-theoretic recurrence theorems require a sharpening of existing tools for handling polynomial multiple ergodic averages. A key advancement made in this paper is a new result on the equidistribution of polynomial orbits in nilmanifolds, which can be seen as a far-reaching generalisation of Weyl’s equidistribution theorem for polynomials of several variables: (3) let $d, k, l in mathbb{N}$ . Let $(X, mathcal{B}, mu, T_1, dots, T_l)$ be an ergodic, connected $mathbb{Z}^l$ -nilsystem. Let ${p_{i,j} ;:; 1 le i le k, 1 le j le l} subseteq mathbb{Q}[x_1, dots, x_d]$ be a family of polynomials such that $p_{i,j}left( mathbb{Z}^d right) subseteq mathbb{Z}$ and ${1} cup {p_{i,j}}$ is linearly independent over $mathbb{Q}$ . Then the $mathbb{Z}^d$ -sequence $left( prod_{j=1}^l{T_j^{p_{1,j}(n)}}x, dots, prod_{j=1}^l{T_j^{p_{k,j}(n)}}x right)_{n in mathbb{Z}^d}$ is well-distributed in $X^k$ for every x in a co-meager set of full measure.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"304 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135830931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Lower bounds on the maximal number of rational points on curves over finite fields","authors":"Jonas Bergström, Everett W. Howe, Elisa Lorenzo García, Christophe Ritzenthaler","doi":"10.1017/s0305004123000476","DOIUrl":"https://doi.org/10.1017/s0305004123000476","url":null,"abstract":"Abstract For a given genus $g geq 1$ , we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $mathbb{F}_q$ . As a consequence of Katz–Sarnak theory, we first get for any given $g>0$ , any $varepsilon>0$ and all q large enough, the existence of a curve of genus g over $mathbb{F}_q$ with at least $1+q+ (2g-varepsilon) sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 sqrt{q}$ valid for $g geq 3$ and odd $q geq 11$ . Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 sqrt{q} -32$ is valid for all $gge 2$ and for all q .","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135343288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Categories of graphs for operadic structures","authors":"Philip Hackney","doi":"10.1017/s0305004123000452","DOIUrl":"https://doi.org/10.1017/s0305004123000452","url":null,"abstract":"Abstract We recall several categories of graphs which are useful for describing homotopy-coherent versions of generalised operads (e.g. cyclic operads, modular operads, properads, and so on), and give new, uniform definitions for their morphisms. This allows for straightforward comparisons, and we use this to show that certain free-forgetful adjunctions between categories of generalised operads can be realised at the level of presheaves. This includes adjunctions between operads and cyclic operads, between dioperads and augmented cyclic operads, and between wheeled properads and modular operads.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135342994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-Orientable Lagrangian Fillings of Legendrian Knots","authors":"LINYI CHEN, GRANT CRIDER-PHILLIPS, BRAEDEN REINOSO, JOSHUA SABLOFF, LEYU YAO","doi":"10.1017/s0305004123000440","DOIUrl":"https://doi.org/10.1017/s0305004123000440","url":null,"abstract":"Abstract We investigate when a Legendrian knot in the standard contact ${{mathbb{R}}}^3$ has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability, and determine when several families of knots have such fillings. In particular, we completely determine when an alternating knot (and more generally a plus-adequate knot) is decomposably non-orientably fillable and classify the fillability of most torus and 3-strand pretzel knots. We also describe rigidity phenomena of decomposable non-orientable fillings, including finiteness of the possible normal Euler numbers of fillings and the minimisation of crosscap numbers of fillings, obtaining results which contrast in interesting ways with the smooth setting.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"326 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135539144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dirichlet law for factorisation of integers, polynomials and permutations","authors":"Sun-Kai Leung","doi":"10.1017/S0305004123000427","DOIUrl":"https://doi.org/10.1017/S0305004123000427","url":null,"abstract":"Abstract Let \u0000$k geqslant 2$\u0000 be an integer. We prove that factorisation of integers into k parts follows the Dirichlet distribution \u0000$mathrm{Dir}left({1}/{k},ldots,{1}/{k}right)$\u0000 by multidimensional contour integration, thereby generalising the Deshouillers–Dress–Tenenbaum (DDT) arcsine law on divisors where \u0000$k=2$\u0000 . The same holds for factorisation of polynomials or permutations. Dirichlet distribution with arbitrary parameters can be modelled similarly.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"1 1","pages":"649 - 676"},"PeriodicalIF":0.8,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89251839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"PSP volume 175 issue 2 Cover and Back matter","authors":"","doi":"10.1017/s0305004123000397","DOIUrl":"https://doi.org/10.1017/s0305004123000397","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"34 1","pages":"b1 - b2"},"PeriodicalIF":0.8,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89919205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"PSP volume 175 issue 2 Cover and Front matter","authors":"","doi":"10.1017/s0305004123000385","DOIUrl":"https://doi.org/10.1017/s0305004123000385","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"569 1","pages":"f1 - f2"},"PeriodicalIF":0.8,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91373969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Prime divisors and the number of conjugacy classes of finite groups","authors":"THOMAS MICHAEL KELLER, ALEXANDER MORETÓ","doi":"10.1017/s030500412300035x","DOIUrl":"https://doi.org/10.1017/s030500412300035x","url":null,"abstract":"We prove that there exists a universal constant <jats:italic>D</jats:italic> such that if <jats:italic>p</jats:italic> is a prime divisor of the index of the Fitting subgroup of a finite group <jats:italic>G</jats:italic>, then the number of conjugacy classes of <jats:italic>G</jats:italic> is at least <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030500412300035X_inline1.png\" /> <jats:tex-math> $Dp/log_2p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We conjecture that we can take <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030500412300035X_inline2.png\" /> <jats:tex-math> $D=1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and prove that for solvable groups, we can take <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030500412300035X_inline3.png\" /> <jats:tex-math> $D=1/3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"14 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"PSP volume 175 issue 1 Cover and Back matter","authors":"","doi":"10.1017/s0305004123000312","DOIUrl":"https://doi.org/10.1017/s0305004123000312","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"263 1","pages":"b1 - b2"},"PeriodicalIF":0.8,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75771505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}