We consider the problem of estimating the mean of a random vector based on $N$ independent, identically distributed observations. We prove the existence of an estimator that has a near-optimal error in all directions in which the variance of the one dimensional marginal of the random vector is not too small: with probability $1-delta$, the procedure returns $wh{mu}_N$ which satisfies that for every direction $u in S^{d-1}$, [ inr{wh{mu}_N - mu, u}le frac{C}{sqrt{N}} left( sigma(u)sqrt{log(1/delta)} + left(E|X-EXP X|_2^2right)^{1/2} right)~, ] where $sigma^2(u) = var(inr{X,u})$ and $C$ is a constant. To achieve this, we require only slightly more than the existence of the covariance matrix, in the form of a certain moment-equivalence assumption. The proof relies on novel bounds for the ratio of empirical and true probabilities that hold uniformly over certain classes of random variables.
{"title":"Multivariate mean estimation with direction-dependent accuracy","authors":"G. Lugosi, S. Mendelson","doi":"10.4171/jems/1321","DOIUrl":"https://doi.org/10.4171/jems/1321","url":null,"abstract":"We consider the problem of estimating the mean of a random vector based on $N$ independent, identically distributed observations. We prove the existence of an estimator that has a near-optimal error in all directions in which the variance of the one dimensional marginal of the random vector is not too small: with probability $1-delta$, the procedure returns $wh{mu}_N$ which satisfies that for every direction $u in S^{d-1}$, [ inr{wh{mu}_N - mu, u}le frac{C}{sqrt{N}} left( sigma(u)sqrt{log(1/delta)} + left(E|X-EXP X|_2^2right)^{1/2} right)~, ] where $sigma^2(u) = var(inr{X,u})$ and $C$ is a constant. To achieve this, we require only slightly more than the existence of the covariance matrix, in the form of a certain moment-equivalence assumption. \u0000The proof relies on novel bounds for the ratio of empirical and true probabilities that hold uniformly over certain classes of random variables.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"51 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85579588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
B. Adamczewski, T. Dreyfus, C. Hardouin, M. Wibmer
We consider pairs of automorphisms $(phi,sigma)$ acting on fields of Laurent or Puiseux series: pairs of shift operators $(phicolon xmapsto x+h_1, sigmacolon xmapsto x+h_2)$, of $q$-difference operators $(phicolon xmapsto q_1x, sigmacolon xmapsto q_2x)$, and of Mahler operators $(phicolon xmapsto x^{p_1}, sigmacolon xmapsto x^{p_2})$. Given a solution $f$ to a linear $phi$-equation and a solution $g$ to a linear $sigma$-equation, both transcendental, we show that $f$ and $g$ are algebraically independent over the field of rational functions, assuming that the corresponding parameters are sufficiently independent. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of $q$-hypergeometric functions. Our approach provides a general strategy to study this kind of question and is based on a suitable Galois theory: the $sigma$-Galois theory of linear $phi$-equations.
{"title":"Algebraic independence and linear difference equations","authors":"B. Adamczewski, T. Dreyfus, C. Hardouin, M. Wibmer","doi":"10.4171/jems/1316","DOIUrl":"https://doi.org/10.4171/jems/1316","url":null,"abstract":"We consider pairs of automorphisms $(phi,sigma)$ acting on fields of Laurent or Puiseux series: pairs of shift operators $(phicolon xmapsto x+h_1, sigmacolon xmapsto x+h_2)$, of $q$-difference operators $(phicolon xmapsto q_1x, sigmacolon xmapsto q_2x)$, and of Mahler operators $(phicolon xmapsto x^{p_1}, sigmacolon xmapsto x^{p_2})$. Given a solution $f$ to a linear $phi$-equation and a solution $g$ to a linear $sigma$-equation, both transcendental, we show that $f$ and $g$ are algebraically independent over the field of rational functions, assuming that the corresponding parameters are sufficiently independent. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of $q$-hypergeometric functions. Our approach provides a general strategy to study this kind of question and is based on a suitable Galois theory: the $sigma$-Galois theory of linear $phi$-equations.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"16 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84500967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study the log-log blowup dynamics for the mass critical nonlinear Schrodinger equation on $mathbb{R}^{2}$ under rough but structured random perturbations at $L^{2}(mathbb{R}^2)$ regularity. In particular, by employing probabilistic methods, we provide a construction of a family of $L^{2}(mathbb{R}^2)$ regularity solutions which do not lie in any $H^{s}(mathbb{R}^2)$ for any $s>0$, and which blowup according to the log-log dynamics.
{"title":"Construction of $L^2$ log-log blowup solutions for the mass critical nonlinear Schrödinger equation","authors":"Chenjie Fan, Dana Mendelson","doi":"10.4171/jems/1314","DOIUrl":"https://doi.org/10.4171/jems/1314","url":null,"abstract":"In this article, we study the log-log blowup dynamics for the mass critical nonlinear Schrodinger equation on $mathbb{R}^{2}$ under rough but structured random perturbations at $L^{2}(mathbb{R}^2)$ regularity. In particular, by employing probabilistic methods, we provide a construction of a family of $L^{2}(mathbb{R}^2)$ regularity solutions which do not lie in any $H^{s}(mathbb{R}^2)$ for any $s>0$, and which blowup according to the log-log dynamics.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"20 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75298898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the truncated correlation functions of the charge and gradient fields associated with the massless sine-Gordon model on $mathbb{R}^2$ with $beta=4pi$ exist for all coupling constants and are equal to those of the chiral densities and vector current of free massive Dirac fermions. This is an instance of Coleman's prediction that the massless sine-Gordon model and the massive Thirring model are equivalent (in the above sense of correlation functions). Our main novelty is that we prove this correspondence in the non-perturative regime of the infinite volume models. We use this correspondence to show that the correlation functions of the massless sine-Gordon model with $beta=4pi$ decay exponentially and that the corresponding probabilistic field is localized.
{"title":"The Coleman correspondence at the free fermion point","authors":"R. Bauerschmidt, Christian Webb","doi":"10.4171/JEMS/1329","DOIUrl":"https://doi.org/10.4171/JEMS/1329","url":null,"abstract":"We prove that the truncated correlation functions of the charge and gradient fields associated with the massless sine-Gordon model on $mathbb{R}^2$ with $beta=4pi$ exist for all coupling constants and are equal to those of the chiral densities and vector current of free massive Dirac fermions. This is an instance of Coleman's prediction that the massless sine-Gordon model and the massive Thirring model are equivalent (in the above sense of correlation functions). Our main novelty is that we prove this correspondence in the non-perturative regime of the infinite volume models. We use this correspondence to show that the correlation functions of the massless sine-Gordon model with $beta=4pi$ decay exponentially and that the corresponding probabilistic field is localized.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"34 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78604397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to the local null-controllability of the nonlinear KdV equation equipped the Dirichlet boundary conditions using the Neumann boundary control on the right. Rosier proved that this KdV system is small-time locally controllable for all non-critical lengths and that the uncontrollable space of the linearized system is of finite dimension when the length is critical. Concerning critical lengths, Coron and Cr'{e}peau showed that the same result holds when the uncontrollable space of the linearized system is of dimension 1, and later Cerpa, and then Cerpa and Cr'epeau established that the local controllability holds at a finite time for all other critical lengths. In this paper, we prove that, for a class of critical lengths, the nonlinear KdV system is {it not} small-time locally controllable.
{"title":"On the small-time local controllability of a KdV system for critical lengths","authors":"J. Coron, Armand Koenig, Hoai-Minh Nguyen","doi":"10.4171/jems/1307","DOIUrl":"https://doi.org/10.4171/jems/1307","url":null,"abstract":"This paper is devoted to the local null-controllability of the nonlinear KdV equation equipped the Dirichlet boundary conditions using the Neumann boundary control on the right. Rosier proved that this KdV system is small-time locally controllable for all non-critical lengths and that the uncontrollable space of the linearized system is of finite dimension when the length is critical. Concerning critical lengths, Coron and Cr'{e}peau showed that the same result holds when the uncontrollable space of the linearized system is of dimension 1, and later Cerpa, and then Cerpa and Cr'epeau established that the local controllability holds at a finite time for all other critical lengths. In this paper, we prove that, for a class of critical lengths, the nonlinear KdV system is {it not} small-time locally controllable.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"38 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90516964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the Holmes--Thompson area of every Finsler disk of radius $r$ whose interior geodesics are length-minimizing is at least $frac{6}{pi} r^2$. Furthermore, we construct examples showing that the inequality is sharp and observe that the equality case is attained by a non-rotationally symmetric metric. This contrasts with Berger's conjecture in the Riemannian case, which asserts that the round hemisphere is extremal. To prove our theorem we discretize the Finsler metric using random geodesics. As an auxiliary result, we show that the integral geometry formulas of Blaschke and Santal'o hold on Finsler manifolds with almost no trapped geodesics.
{"title":"Minimal area of Finsler disks with minimizing geodesics","authors":"Marcos Cossarini, S. Sabourau","doi":"10.4171/JEMS/1339","DOIUrl":"https://doi.org/10.4171/JEMS/1339","url":null,"abstract":"We show that the Holmes--Thompson area of every Finsler disk of radius $r$ whose interior geodesics are length-minimizing is at least $frac{6}{pi} r^2$. Furthermore, we construct examples showing that the inequality is sharp and observe that the equality case is attained by a non-rotationally symmetric metric. This contrasts with Berger's conjecture in the Riemannian case, which asserts that the round hemisphere is extremal. To prove our theorem we discretize the Finsler metric using random geodesics. As an auxiliary result, we show that the integral geometry formulas of Blaschke and Santal'o hold on Finsler manifolds with almost no trapped geodesics.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"537 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77451811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given an ample real Hermitian holomorphic line bundle $L$ over a real algebraic variety $X$, the space of real holomorphic sections of $L^{otimes d}$ inherits a natural Gaussian probability measure. We prove that the probability that the zero locus of a real holomorphic section $s$ of $L^{otimes d}$ defines a maximal hypersurface tends to $0$ exponentially fast as $d$ goes to infinity. This extends to any dimension a result of Gayet and Welschinger valid for maximal real algebraic curves inside a real algebraic surface. The starting point is a low degree approximation property which relates the topology of the real vanishing locus of a real holomorphic section of $L^{otimes d}$ with the topology of the real vanishing locus a real holomorphic section of $L^{otimes d'}$ for a sufficiently smaller $d'
{"title":"Exponential rarefaction of maximal real algebraic hypersurfaces","authors":"Michele Ancona","doi":"10.4171/jems/1311","DOIUrl":"https://doi.org/10.4171/jems/1311","url":null,"abstract":"Given an ample real Hermitian holomorphic line bundle $L$ over a real algebraic variety $X$, the space of real holomorphic sections of $L^{otimes d}$ inherits a natural Gaussian probability measure. We prove that the probability that the zero locus of a real holomorphic section $s$ of $L^{otimes d}$ defines a maximal hypersurface tends to $0$ exponentially fast as $d$ goes to infinity. This extends to any dimension a result of Gayet and Welschinger valid for maximal real algebraic curves inside a real algebraic surface. \u0000The starting point is a low degree approximation property which relates the topology of the real vanishing locus of a real holomorphic section of $L^{otimes d}$ with the topology of the real vanishing locus a real holomorphic section of $L^{otimes d'}$ for a sufficiently smaller $d'","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"40 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83023298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Suppose that $M$ is a closed isotropic Riemannian manifold and that $R_1,...,R_m$ generate the isometry group of $M$. Let $f_1,...,f_m$ be smooth perturbations of these isometries. We show that the $f_i$ are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian from $S^n$ to real, complex, and quaternionic projective spaces. In addition, we identify and remedy an oversight in that earlier work.
{"title":"Simultaneous linearization of diffeomorphisms of isotropic manifolds","authors":"Jonathan DeWitt","doi":"10.4171/jems/1327","DOIUrl":"https://doi.org/10.4171/jems/1327","url":null,"abstract":"Suppose that $M$ is a closed isotropic Riemannian manifold and that $R_1,...,R_m$ generate the isometry group of $M$. Let $f_1,...,f_m$ be smooth perturbations of these isometries. We show that the $f_i$ are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian from $S^n$ to real, complex, and quaternionic projective spaces. In addition, we identify and remedy an oversight in that earlier work.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"15 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79168348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove an $(l^2, l^6)$ decoupling inequality for the parabola with constant $(log R)^c$. In the appendix, we present an application to the six-order correlation of the integer solutions to $x^2+y^2=m$.
{"title":"Improved decoupling for the parabola","authors":"L. Guth, Dominique Maldague, Hong Wang","doi":"10.4171/jems/1295","DOIUrl":"https://doi.org/10.4171/jems/1295","url":null,"abstract":"We prove an $(l^2, l^6)$ decoupling inequality for the parabola with constant $(log R)^c$. In the appendix, we present an application to the six-order correlation of the integer solutions to $x^2+y^2=m$.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"150 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74853794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The embedding of a torus into an inner form of PGL(2) defines an adelic toric period. A general version of Duke's theorem states that this period equidistributes as the discriminant of the splitting field tends to infinity. In this paper we consider a torus embedded diagonally into two distinct inner forms of PGL(2). Assuming the Generalized Riemann Hypothesis (and some additional technical assumptions), we show simultaneous equidistribution as the discriminant tends to infinity, with an effective logarithmic rate. Our proof is based on a probabilistic approach to estimating fractional moments of L-functions twisted by extended class group characters.
{"title":"Simultaneous equidistribution of toric periods and fractional moments of $L$-functions","authors":"V. Blomer, Farrell Brumley","doi":"10.4171/jems/1324","DOIUrl":"https://doi.org/10.4171/jems/1324","url":null,"abstract":"The embedding of a torus into an inner form of PGL(2) defines an adelic toric period. A general version of Duke's theorem states that this period equidistributes as the discriminant of the splitting field tends to infinity. In this paper we consider a torus embedded diagonally into two distinct inner forms of PGL(2). Assuming the Generalized Riemann Hypothesis (and some additional technical assumptions), we show simultaneous equidistribution as the discriminant tends to infinity, with an effective logarithmic rate. Our proof is based on a probabilistic approach to estimating fractional moments of L-functions twisted by extended class group characters.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"93 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83887987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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