Pub Date : 2024-07-17DOI: 10.4310/mrl.2023.v30.n6.a2
Suresh Eswarathasan, Xiaolong Han
In this paper, we investigate the fractal uncertainty principle (FUP) for discrete Cantor sets, which are determined by an alphabet from a base of digits. Consider the base of $M$ digits and the alphabets of cardinality $A$ such that all the corresponding Cantor sets have a fixed dimension $log A/log Min (0,2/3)$. We prove that the FUP with an improved exponent over Dyatlov-Jin $href{https://doi.org/10.48550/arXiv.2107.08276}{textrm{DJ-1}}$ holds for almost all alphabets, asymptotically as $Mtoinfty$. Our result provides the best possible exponent when the Cantor sets enjoy either the strongest Fourier decay assumption or strongest additive energy assumption. The proof is based on a concentration of measure phenomenon in the space of alphabets.
本文研究了离散康托集合的分形不确定性原理（FUP），康托集合是由数字基数的字母表决定的。考虑由 $M$ 数字组成的基数和 cardinality $A$ 的字母表，所有相应的 Cantor 集都有一个固定维度 $log A/log Min (0,2/3)$。我们证明，对于几乎所有的字母集，FUP 的指数都比 Dyatlov-Jin $href{https://doi.org/10.48550/arXiv.2107.08276}{textrm{DJ-1}}$ 高，且渐近于 $Mtoinfty$。当康托集合享有最强傅里叶衰变假设或最强加法能量假设时，我们的结果提供了可能的最佳指数。证明基于字母空间中的度量集中现象。
{"title":"Fractal uncertainty principle for discrete Cantor sets with random alphabets","authors":"Suresh Eswarathasan, Xiaolong Han","doi":"10.4310/mrl.2023.v30.n6.a2","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n6.a2","url":null,"abstract":"In this paper, we investigate the fractal uncertainty principle (FUP) for discrete Cantor sets, which are determined by an alphabet from a base of digits. Consider the base of $M$ digits and the alphabets of cardinality $A$ such that all the corresponding Cantor sets have a fixed dimension $log A/log Min (0,2/3)$. We prove that the FUP with an improved exponent over Dyatlov-Jin $href{https://doi.org/10.48550/arXiv.2107.08276}{textrm{DJ-1}}$ holds for almost all alphabets, asymptotically as $Mtoinfty$. Our result provides the best possible exponent when the Cantor sets enjoy either the strongest Fourier decay assumption or strongest additive energy assumption. The proof is based on a concentration of measure phenomenon in the space of alphabets.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"36 Suppl 2 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744903","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 : 2024-05-14DOI: 10.4310/mrl.2023.v30.n5.a7
Jun Hu, Wei Xiao
In this paper we give a sum formula for the radical filtration of parabolic Verma modules in any (possibly singular) blocks of parabolic BGG category. It can be viewed as a generalization of the Jantzen sum formula for Verma modules in the usual BGG category $mathcal{O}$. The proof makes use of the graded version of parabolic BGG category. Explicit formulae for the graded decomposition numbers and inverse graded decomposition numbers of parabolic Verma modules in any (possibly singular) integral blocks of the parabolic BGG category are also given in terms of the Kazhdan–Lusztig polynomials.
{"title":"On radical filtrations of parabolic Verma modules","authors":"Jun Hu, Wei Xiao","doi":"10.4310/mrl.2023.v30.n5.a7","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n5.a7","url":null,"abstract":"In this paper we give a sum formula for the radical filtration of parabolic Verma modules in any (possibly singular) blocks of parabolic BGG category. It can be viewed as a generalization of the Jantzen sum formula for Verma modules in the usual BGG category $mathcal{O}$. The proof makes use of the graded version of parabolic BGG category. Explicit formulae for the graded decomposition numbers and inverse graded decomposition numbers of parabolic Verma modules in any (possibly singular) integral blocks of the parabolic BGG category are also given in terms of the Kazhdan–Lusztig polynomials.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"47 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941577","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 : 2024-05-14DOI: 10.4310/mrl.2023.v30.n5.a1
Hajer Bahouri, Trevor M. Leslie, Galina Perelman
We study the derivative nonlinear Schrödinger equation on the real line and obtain global-in-time bounds on high order Sobolev norms.
我们研究了实线上的导数非线性薛定谔方程，并获得了高阶索波列夫规范的全局时间界限。
{"title":"$H^s$ bounds for the derivative nonlinear Schrödinger equation","authors":"Hajer Bahouri, Trevor M. Leslie, Galina Perelman","doi":"10.4310/mrl.2023.v30.n5.a1","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n5.a1","url":null,"abstract":"We study the derivative nonlinear Schrödinger equation on the real line and obtain global-in-time bounds on high order Sobolev norms.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"27 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941462","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 : 2024-05-14DOI: 10.4310/mrl.2023.v30.n5.a13
Ferenc Szöllősi
The kissing number $tau (d)$ is the maximum number of pairwise non-overlapping unit spheres each touching a central unit sphere in the $d$-dimensional Euclidean space. In this note we report on how we discovered a new, previously unknown arrangement of 40 unit spheres in dimension $5$. Our arrangement saturates the best known lower bound on $tau (5)$, and refutes a ‘belief’ of Cohn–Jiao–Kumar–Torquato.
{"title":"A note on five dimensional kissing arrangements","authors":"Ferenc Szöllősi","doi":"10.4310/mrl.2023.v30.n5.a13","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n5.a13","url":null,"abstract":"The kissing number $tau (d)$ is the maximum number of pairwise non-overlapping unit spheres each touching a central unit sphere in the $d$-dimensional Euclidean space. In this note we report on how we discovered a new, previously unknown arrangement of 40 unit spheres in dimension $5$. Our arrangement saturates the best known lower bound on $tau (5)$, and refutes a ‘belief’ of Cohn–Jiao–Kumar–Torquato.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"77 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941576","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 : 2024-05-14DOI: 10.4310/mrl.2023.v30.n5.a9
Ariyan Javanpeykar
Motivated by the intermediate Lang conjectures on hyperbolicity and rational points, we prove new finiteness results for non-constant morphisms from a fixed variety to a fixed variety defined over a number field by applying Faltings’s finiteness results to moduli spaces of maps.
{"title":"Finiteness of non-constant maps over a number field","authors":"Ariyan Javanpeykar","doi":"10.4310/mrl.2023.v30.n5.a9","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n5.a9","url":null,"abstract":"Motivated by the intermediate Lang conjectures on hyperbolicity and rational points, we prove new finiteness results for non-constant morphisms from a fixed variety to a fixed variety defined over a number field by applying Faltings’s finiteness results to moduli spaces of maps.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"19 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941585","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 : 2024-05-14DOI: 10.4310/mrl.2023.v30.n5.a10
Ru-Yu Lai, Ting Zhou
We prove that the knowledge of the Dirichlet-to-Neumann map, measured on a part of the boundary of a bounded domain in $mathbb{R}^n , n geq 2$, can uniquely determine, in a nonlinear magnetic Schrödinger equation, the vector-valued magnetic potential and the scalar electric potential, both being nonlinear in the solution.
我们证明，在$mathbb{R}^n , n geq 2$的有界域的部分边界上测量迪里希勒到诺伊曼映射，可以唯一地确定非线性磁薛定谔方程中的矢量磁势和标量电势，两者在解中都是非线性的。
{"title":"Partial data inverse problems for nonlinear magnetic Schrödinger equations","authors":"Ru-Yu Lai, Ting Zhou","doi":"10.4310/mrl.2023.v30.n5.a10","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n5.a10","url":null,"abstract":"We prove that the knowledge of the Dirichlet-to-Neumann map, measured on a part of the boundary of a bounded domain in $mathbb{R}^n , n geq 2$, can uniquely determine, in a nonlinear magnetic Schrödinger equation, the vector-valued magnetic potential and the scalar electric potential, both being nonlinear in the solution.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"25 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941504","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 : 2024-05-14DOI: 10.4310/mrl.2023.v30.n5.a3
Simon Felten, Andrea Petracci, Sharon Robins
We exhibit examples of pairs $(X,D)$ where $X$ is a smooth projective variety and $D$ is an anticanonical reduced simple normal crossing divisor such that the deformations of $(X,D)$ are obstructed. These examples are constructed via toric geometry.
{"title":"Deformations of $log$ Calabi–Yau pairs can be obstructed","authors":"Simon Felten, Andrea Petracci, Sharon Robins","doi":"10.4310/mrl.2023.v30.n5.a3","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n5.a3","url":null,"abstract":"We exhibit examples of pairs $(X,D)$ where $X$ is a smooth projective variety and $D$ is an anticanonical reduced simple normal crossing divisor such that the deformations of $(X,D)$ are obstructed. These examples are constructed via toric geometry.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"65 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941580","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 : 2024-05-14DOI: 10.4310/mrl.2023.v30.n5.a4
Shaoming Guo, Zane Kun Li, Po-Lam Yung
Using ideas from $href{https://doi.org/10.4171/jems/1295}{[7]}$ and working over $mathbb{Q}_p$, we show that the discrete restriction constant for the parabola is $O_varepsilon ((log M)^{2+varepsilon})$.
{"title":"Improved discrete restriction for the parabola","authors":"Shaoming Guo, Zane Kun Li, Po-Lam Yung","doi":"10.4310/mrl.2023.v30.n5.a4","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n5.a4","url":null,"abstract":"Using ideas from $href{https://doi.org/10.4171/jems/1295}{[7]}$ and working over $mathbb{Q}_p$, we show that the discrete restriction constant for the parabola is $O_varepsilon ((log M)^{2+varepsilon})$.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"22 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941574","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 : 2024-05-14DOI: 10.4310/mrl.2023.v30.n5.a6
Serin Hong
Given two arbitrary vector bundles on the Fargues–Fontaine curve, we give an explicit criterion in terms of Harder–Narasimhan polygons on whether they realize a semistable vector bundle as their extensions. Our argument is largely combinatorial and builds upon the dimension analysis of certain moduli spaces of bundle maps developed in $href{https://doi.org/10.1017/S1474748020000183}{[1]}$.
{"title":"On certain extensions of vector bundles in $p$-adic geometry","authors":"Serin Hong","doi":"10.4310/mrl.2023.v30.n5.a6","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n5.a6","url":null,"abstract":"Given two arbitrary vector bundles on the Fargues–Fontaine curve, we give an explicit criterion in terms of Harder–Narasimhan polygons on whether they realize a semistable vector bundle as their extensions. Our argument is largely combinatorial and builds upon the dimension analysis of certain moduli spaces of bundle maps developed in $href{https://doi.org/10.1017/S1474748020000183}{[1]}$.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"28 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941813","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 : 2024-05-14DOI: 10.4310/mrl.2023.v30.n5.a8
Marius Huber
We show that the notion of ribbon rational homology cobordism yields a partial order on the set of aspherical 3‑manifolds, thus supporting a conjecture formulated by Daemi, Lidman, Vela–Vick and Wong. Our proof is built on Agol’s recent proof of the fact that ribbon concordance yields a partial order on the set of knots in the 3‑sphere.
我们证明了带状理性同调概念在非球面 3-manifolds集合上产生了一个偏序，从而支持了由 Daemi、Lidman、Vela-Vick 和 Wong 提出的猜想。我们的证明建立在阿戈尔最近对带状同调产生 3 球中结集的偏序这一事实的证明之上。
{"title":"Ribbon cobordisms as a partial order","authors":"Marius Huber","doi":"10.4310/mrl.2023.v30.n5.a8","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n5.a8","url":null,"abstract":"We show that the notion of ribbon rational homology cobordism yields a partial order on the set of aspherical 3‑manifolds, thus supporting a conjecture formulated by Daemi, Lidman, Vela–Vick and Wong. Our proof is built on Agol’s recent proof of the fact that ribbon concordance yields a partial order on the set of knots in the 3‑sphere.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"41 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941455","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：