Pub Date : 2021-12-22DOI: 10.1017/S0305004123000373
Annina Iseli, Anton Lukyanenko
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 $min(1, dim A)$ . 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 $PSL(2,mathbb{C})$ or a generic real linear-fractional transformation in $PGL(3,mathbb{R})$ . We next show that, under some necessary technical assumptions, transversality locally holds for restricted families of projections corresponding to one-dimensional subgroups of $PSL(2,mathbb{C})$ or $PGL(3,mathbb{R})$ . 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.
Marstrand定理指出，在平面集合a与x轴正交投影之前，对其进行一般旋转，几乎肯定会得到具有最大可能维数$min(1， dim a)$的图像。我们首先利用Peres-Schlag的局部横向性理论，证明了在$PSL(2，mathbb{C})$中应用一般复线性分数变换或在$PGL(3，mathbb{R})$中应用一般实线性分数变换具有相同的结果。我们接下来证明，在一些必要的技术假设下，对于$PSL(2，mathbb{C})$或$PGL(3，mathbb{R})$的一维子群所对应的有限投影族，横向性局部成立。第三，我们证明了双曲几何和球面几何的全测地线子空间的最近点投影族在任何维上的局部截线性和由此产生的投影命题。
{"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}
Pub Date : 2021-12-20DOI: 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}
Pub Date : 2021-12-20DOI: 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}
Pub Date : 2021-12-14DOI: 10.1017/S0305004121000700
T. Meyrath, J. Müller
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.
{"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}
Pub Date : 2021-12-10DOI: 10.1017/s0305004123000361
Th'eo Untrau
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.
{"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}
Pub Date : 2021-12-03DOI: 10.1017/S0305004123000117
Danilo Lewa'nski
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.
{"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}
Pub Date : 2021-12-03DOI: 10.1017/S0305004122000093
M. Erraoui, Youssef Hakiki
Abstract Let $B^{H}$ be a fractional Brownian motion in $mathbb{R}^{d}$ of Hurst index $Hinleft(0,1right)$ , $f;:;left[0,1right]longrightarrowmathbb{R}^{d}$ a Borel function and $Asubsetleft[0,1right]$ a Borel set. We provide sufficient conditions for the image $(B^{H}+f)(A)$ 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 $(B^{H}+f)$ . 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.
摘要设$B^{H}$是赫斯特指数$H 左(0,1右)$ $中的$mathbb{R}^{d}$ $中的分数布朗运动，$f;:;left[0,1右] longightarrow mathbb{R}^{d}$ a Borel函数和$ a 子集left[0,1右]$ a Borel集合。我们给出了图像$(B^{H}+f)(A)$具有正勒贝格测度或具有非空内部的充分条件。这是通过研究$(B^{H}+f)$的占用测度的密度的性质来实现的。准确地说，我们证明了如果图f的抛物线Hausdorff维数大于Hd，则密度是平方可积函数。另一方面，如果A的Hausdorff维数大于Hd，则它甚至允许存在连续版本。这允许我们建立已经引用的结果。
{"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}
Pub Date : 2021-11-18DOI: 10.1017/S0305004123000221
J. Taylor
Abstract Let F be a finite extension of ${mathbb Q}_p$ . Let $Omega$ be the Drinfeld upper half plane, and $Sigma^1$ the first Drinfeld covering of $Omega$ . We study the affinoid open subset $Sigma^1_v$ of $Sigma^1$ above a vertex of the Bruhat–Tits tree for $text{GL}_2(F)$ . Our main result is that $text{Pic}!left(Sigma^1_vright)[p] = 0$ , which we establish by showing that $text{Pic}({mathbf Y})[p] = 0$ for ${mathbf Y}$ the Deligne–Lusztig variety of $text{SL}_2!left({mathbb F}_qright)$ . One formal consequence is a description of the representation $H^1_{{acute{text{e}}text{t}}}!left(Sigma^1_v, {mathbb Z}_p(1)right)$ of $text{GL}_2(mathcal{O}_F)$ as the p-adic completion of $mathcal{O}!left(Sigma^1_vright)^times$ .
{"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}
Pub Date : 2021-11-17DOI: 10.1017/S0305004123000130
Alex McDonald, K. Taylor
Abstract In this paper we prove that the set ${|x^1-x^2|,dots,|x^k-x^{k+1}|,{:},x^iin E}$ has non-empty interior in $mathbb{R}^k$ when $Esubset mathbb{R}^2$ is a Cartesian product of thick Cantor sets $K_1,K_2subsetmathbb{R}$ . We also prove more general results where the distance map $|x-y|$ is replaced by a function $phi(x,y)$ 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 $K_1,K_2, phi$ are as above then there exists an open set S so that $bigcap_{x in S} phi(x,K_1times K_2)$ has non-empty interior.
摘要本文证明了当$Esubset mathbb{R}^2$是厚康托集$K_1,K_2subsetmathbb{R}$的笛卡尔积时，集合${|x^1-x^2|,dots,|x^k-x^{k+1}|,{:},x^iin E}$在$mathbb{R}^k$中具有非空内。我们还证明了更一般的结果，其中距离图$|x-y|$被一个满足其偏导数温和假设的函数$phi(x,y)$所取代。在此过程中，我们建立了经典Newhouse Gap引理的一个非线性版本，并证明了如果$K_1,K_2, phi$如上所述，则存在一个开集S，使得$bigcap_{x in S} phi(x,K_1times K_2)$具有非空的内部。
{"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}
Pub Date : 2021-11-01DOI: 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}
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