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Projection theorems for linear-fractional families of projections 投影的线性分数族的投影定理
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-12-22 DOI: 10.1017/S0305004123000373
Annina Iseli, Anton Lukyanenko
{"title":"Projection theorems for linear-fractional families of projections","authors":"Annina Iseli, Anton Lukyanenko","doi":"10.1017/S0305004123000373","DOIUrl":"https://doi.org/10.1017/S0305004123000373","url":null,"abstract":"Abstract Marstrand’s theorem states that applying a generic rotation to a planar set A before projecting it orthogonally to the x-axis almost surely gives an image with the maximal possible dimension \u0000$min(1, dim A)$\u0000 . We first prove, using the transversality theory of Peres–Schlag locally, that the same result holds when applying a generic complex linear-fractional transformation in \u0000$PSL(2,mathbb{C})$\u0000 or a generic real linear-fractional transformation in \u0000$PGL(3,mathbb{R})$\u0000 . We next show that, under some necessary technical assumptions, transversality locally holds for restricted families of projections corresponding to one-dimensional subgroups of \u0000$PSL(2,mathbb{C})$\u0000 or \u0000$PGL(3,mathbb{R})$\u0000 . Third, we demonstrate, in any dimension, local transversality and resulting projection statements for the families of closest-point projections to totally-geodesic subspaces of hyperbolic and spherical geometries.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"25 1","pages":"625 - 647"},"PeriodicalIF":0.8,"publicationDate":"2021-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75991727","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}
引用次数: 1
PSP volume 172 issue 1 Cover and Front matter PSP卷172问题1封面和前面的问题
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-12-20 DOI: 10.1017/s0305004121000724
{"title":"PSP volume 172 issue 1 Cover and Front matter","authors":"","doi":"10.1017/s0305004121000724","DOIUrl":"https://doi.org/10.1017/s0305004121000724","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"49 1","pages":"f1 - f2"},"PeriodicalIF":0.8,"publicationDate":"2021-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86632489","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}
引用次数: 0
PSP volume 172 issue 1 Cover and Back matter PSP卷172期1封面和背面
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-12-20 DOI: 10.1017/s0305004121000736
{"title":"PSP volume 172 issue 1 Cover and Back matter","authors":"","doi":"10.1017/s0305004121000736","DOIUrl":"https://doi.org/10.1017/s0305004121000736","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"9 1","pages":"b1 - b3"},"PeriodicalIF":0.8,"publicationDate":"2021-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75295214","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}
引用次数: 0
Non-normality, topological transitivity and expanding families 非正态性，拓扑及性和扩展族
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-12-14 DOI: 10.1017/S0305004121000700
T. Meyrath, J. Müller
{"title":"Non-normality, topological transitivity and expanding families","authors":"T. Meyrath, J. Müller","doi":"10.1017/S0305004121000700","DOIUrl":"https://doi.org/10.1017/S0305004121000700","url":null,"abstract":"Abstract We investigate the behaviour of families of meromorphic functions in the neighbourhood of points of non-normality and prove certain covering properties that complement Montel’s Theorem. In particular, we also obtain characterisations of non-normality in terms of such properties.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"43 1","pages":"511 - 523"},"PeriodicalIF":0.8,"publicationDate":"2021-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87873821","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}
引用次数: 0
Equidistribution of exponential sums indexed by a subgroup of fixed cardinality 由固定基数子群索引的指数和的等分布
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-12-10 DOI: 10.1017/s0305004123000361
Th'eo Untrau
{"title":"Equidistribution of exponential sums indexed by a subgroup of fixed cardinality","authors":"Th'eo Untrau","doi":"10.1017/s0305004123000361","DOIUrl":"https://doi.org/10.1017/s0305004123000361","url":null,"abstract":"\u0000 We consider families of exponential sums indexed by a subgroup of invertible classes modulo some prime power q. For fixed d, we restrict to moduli q so that there is a unique subgroup of invertible classes modulo q of order d. We study distribution properties of these families of sums as q grows and we establish equidistribution results in some regions of the complex plane which are described as the image of a multi-dimensional torus via an explicit Laurent polynomial. In some cases, the region of equidistribution can be interpreted as the one delimited by a hypocycloid, or as a Minkowski sum of such regions.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"17 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84696470","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}
引用次数: 2
On some hyperelliptic Hurwitz–Hodge integrals 关于一些超椭圆型Hurwitz-Hodge积分
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-12-03 DOI: 10.1017/S0305004123000117
Danilo Lewa'nski
{"title":"On some hyperelliptic Hurwitz–Hodge integrals","authors":"Danilo Lewa'nski","doi":"10.1017/S0305004123000117","DOIUrl":"https://doi.org/10.1017/S0305004123000117","url":null,"abstract":"Abstract We address Hodge integrals over the hyperelliptic locus. Recently Afandi computed, via localisation techniques, such one-descendant integrals and showed that they are Stirling numbers. We give another proof of the same statement by a very short argument, exploiting Chern classes of spin structures and relations arising from Topological Recursion in the sense of Eynard and Orantin. These techniques seem also suitable to deal with three orthogonal generalisations: (1) the extension to the r-hyperelliptic locus; (2) the extension to an arbitrary number of non-Weierstrass pairs of points; (3) the extension to multiple descendants.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"17 1","pages":"271 - 284"},"PeriodicalIF":0.8,"publicationDate":"2021-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82057114","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}
引用次数: 1
Images of fractional Brownian motion with deterministic drift: Positive Lebesgue measure and non-empty interior 具有确定性漂移的分数布朗运动图像:正勒贝格测度和非空内部
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-12-03 DOI: 10.1017/S0305004122000093
M. Erraoui, Youssef Hakiki
{"title":"Images of fractional Brownian motion with deterministic drift: Positive Lebesgue measure and non-empty interior","authors":"M. Erraoui, Youssef Hakiki","doi":"10.1017/S0305004122000093","DOIUrl":"https://doi.org/10.1017/S0305004122000093","url":null,"abstract":"Abstract Let \u0000$B^{H}$\u0000 be a fractional Brownian motion in \u0000$mathbb{R}^{d}$\u0000 of Hurst index \u0000$Hinleft(0,1right)$\u0000 , \u0000$f;:;left[0,1right]longrightarrowmathbb{R}^{d}$\u0000 a Borel function and \u0000$Asubsetleft[0,1right]$\u0000 a Borel set. We provide sufficient conditions for the image \u0000$(B^{H}+f)(A)$\u0000 to have a positive Lebesgue measure or to have a non-empty interior. This is done through the study of the properties of the density of the occupation measure of \u0000$(B^{H}+f)$\u0000 . Precisely, we prove that if the parabolic Hausdorff dimension of the graph of f is greater than Hd, then the density is a square integrable function. If, on the other hand, the Hausdorff dimension of A is greater than Hd, then it even admits a continuous version. This allows us to establish the result already cited.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"6 1","pages":"693 - 713"},"PeriodicalIF":0.8,"publicationDate":"2021-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82015627","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}
引用次数: 4
The Picard group of vertex affinoids in the first Drinfeld covering 第一Drinfeld覆盖中的Picard顶点仿射群
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-11-18 DOI: 10.1017/S0305004123000221
J. Taylor
{"title":"The Picard group of vertex affinoids in the first Drinfeld covering","authors":"J. Taylor","doi":"10.1017/S0305004123000221","DOIUrl":"https://doi.org/10.1017/S0305004123000221","url":null,"abstract":"Abstract Let F be a finite extension of \u0000${mathbb Q}_p$\u0000 . Let \u0000$Omega$\u0000 be the Drinfeld upper half plane, and \u0000$Sigma^1$\u0000 the first Drinfeld covering of \u0000$Omega$\u0000 . We study the affinoid open subset \u0000$Sigma^1_v$\u0000 of \u0000$Sigma^1$\u0000 above a vertex of the Bruhat–Tits tree for \u0000$text{GL}_2(F)$\u0000 . Our main result is that \u0000$text{Pic}!left(Sigma^1_vright)[p] = 0$\u0000 , which we establish by showing that \u0000$text{Pic}({mathbf Y})[p] = 0$\u0000 for \u0000${mathbf Y}$\u0000 the Deligne–Lusztig variety of \u0000$text{SL}_2!left({mathbb F}_qright)$\u0000 . One formal consequence is a description of the representation \u0000$H^1_{{acute{text{e}}text{t}}}!left(Sigma^1_v, {mathbb Z}_p(1)right)$\u0000 of \u0000$text{GL}_2(mathcal{O}_F)$\u0000 as the p-adic completion of \u0000$mathcal{O}!left(Sigma^1_vright)^times$\u0000 .","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"112 1","pages":"423 - 432"},"PeriodicalIF":0.8,"publicationDate":"2021-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78247657","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}
引用次数: 2
Finite point configurations in products of thick Cantor sets and a robust nonlinear Newhouse Gap Lemma 厚Cantor集积的有限点构型及鲁棒非线性Newhouse Gap引理
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-11-17 DOI: 10.1017/S0305004123000130
Alex McDonald, K. Taylor
{"title":"Finite point configurations in products of thick Cantor sets and a robust nonlinear Newhouse Gap Lemma","authors":"Alex McDonald, K. Taylor","doi":"10.1017/S0305004123000130","DOIUrl":"https://doi.org/10.1017/S0305004123000130","url":null,"abstract":"Abstract In this paper we prove that the set \u0000${|x^1-x^2|,dots,|x^k-x^{k+1}|,{:},x^iin E}$\u0000 has non-empty interior in \u0000$mathbb{R}^k$\u0000 when \u0000$Esubset mathbb{R}^2$\u0000 is a Cartesian product of thick Cantor sets \u0000$K_1,K_2subsetmathbb{R}$\u0000 . We also prove more general results where the distance map \u0000$|x-y|$\u0000 is replaced by a function \u0000$phi(x,y)$\u0000 satisfying mild assumptions on its partial derivatives. In the process, we establish a nonlinear version of the classic Newhouse Gap Lemma, and show that if \u0000$K_1,K_2, phi$\u0000 are as above then there exists an open set S so that \u0000$bigcap_{x in S} phi(x,K_1times K_2)$\u0000 has non-empty interior.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"46 1","pages":"285 - 301"},"PeriodicalIF":0.8,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78929502","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}
引用次数: 11
PSP volume 171 issue 3 Cover and Back matter PSP第171卷第3期封面和封底
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-11-01 DOI: 10.1017/s0305004121000621
{"title":"PSP volume 171 issue 3 Cover and Back matter","authors":"","doi":"10.1017/s0305004121000621","DOIUrl":"https://doi.org/10.1017/s0305004121000621","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"72 1","pages":"b1 - b2"},"PeriodicalIF":0.8,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84160319","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}
引用次数: 0
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