{"title":"Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages","authors":"Zemer Kosloff, Shrey Sanadhya","doi":"arxiv-2409.05087","DOIUrl":null,"url":null,"abstract":"We show that for every ergodic and aperiodic probability preserving system\n$(X,\\mathcal{B},m,T)$, there exists $f:X\\to \\mathbb{Z}^d$, whose corresponding\ncocycle satisfies the d-dimensional local central limit theorem. We use the 2-dimensional result to resolve a question of Huang, Shao and Ye\nand Franzikinakis and Host regarding non-convergence of polynomial multiple\naverages of non-commuting zero entropy transformations.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"8 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05087","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We show that for every ergodic and aperiodic probability preserving system
$(X,\mathcal{B},m,T)$, there exists $f:X\to \mathbb{Z}^d$, whose corresponding
cocycle satisfies the d-dimensional local central limit theorem. We use the 2-dimensional result to resolve a question of Huang, Shao and Ye
and Franzikinakis and Host regarding non-convergence of polynomial multiple
averages of non-commuting zero entropy transformations.