{"title":"无l形构型的子集","authors":"Sarah Peluse","doi":"10.1112/s0010437x2300756x","DOIUrl":null,"url":null,"abstract":"<p>Fix a prime <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$p\\geq 11$</span></span></img></span></span>. We show that there exists a positive integer <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$m$</span></span></img></span></span> such that any subset of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbb {F}_p^n\\times \\mathbb {F}_p^n$</span></span></img></span></span> containing no nontrivial configurations of the form <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$(x,y)$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$(x,y+z)$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$(x,y+2z)$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$(x+z,y)$</span></span></img></span></span> must have density <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline11.png\"><span data-mathjax-type=\"texmath\"><span>$\\ll 1/\\log _{m}{n}$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline12.png\"><span data-mathjax-type=\"texmath\"><span>$\\log _{m}$</span></span></img></span></span> denotes the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline13.png\"><span data-mathjax-type=\"texmath\"><span>$m$</span></span></img></span></span>-fold iterated logarithm. This gives the first reasonable bound in the multidimensional Szemerédi theorem for a two-dimensional four-point configuration in any setting.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"20 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Subsets of without L-shaped configurations\",\"authors\":\"Sarah Peluse\",\"doi\":\"10.1112/s0010437x2300756x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Fix a prime <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline3.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$p\\\\geq 11$</span></span></img></span></span>. We show that there exists a positive integer <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline4.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$m$</span></span></img></span></span> such that any subset of <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline5.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathbb {F}_p^n\\\\times \\\\mathbb {F}_p^n$</span></span></img></span></span> containing no nontrivial configurations of the form <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline7.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$(x,y)$</span></span></img></span></span>, <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline8.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$(x,y+z)$</span></span></img></span></span>, <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline9.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$(x,y+2z)$</span></span></img></span></span>, <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline10.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$(x+z,y)$</span></span></img></span></span> must have density <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline11.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\ll 1/\\\\log _{m}{n}$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline12.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\log _{m}$</span></span></img></span></span> denotes the <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline13.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$m$</span></span></img></span></span>-fold iterated logarithm. This gives the first reasonable bound in the multidimensional Szemerédi theorem for a two-dimensional four-point configuration in any setting.</p>\",\"PeriodicalId\":55232,\"journal\":{\"name\":\"Compositio Mathematica\",\"volume\":\"20 1\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2023-12-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Compositio Mathematica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1112/s0010437x2300756x\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Compositio Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/s0010437x2300756x","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Fix a prime $p\geq 11$. We show that there exists a positive integer $m$ such that any subset of $\mathbb {F}_p^n\times \mathbb {F}_p^n$ containing no nontrivial configurations of the form $(x,y)$, $(x,y+z)$, $(x,y+2z)$, $(x+z,y)$ must have density $\ll 1/\log _{m}{n}$, where $\log _{m}$ denotes the $m$-fold iterated logarithm. This gives the first reasonable bound in the multidimensional Szemerédi theorem for a two-dimensional four-point configuration in any setting.
期刊介绍:
Compositio Mathematica is a prestigious, well-established journal publishing first-class research papers that traditionally focus on the mainstream of pure mathematics. Compositio Mathematica has a broad scope which includes the fields of algebra, number theory, topology, algebraic and differential geometry and global analysis. Papers on other topics are welcome if they are of broad interest. All contributions are required to meet high standards of quality and originality. The Journal has an international editorial board reflected in the journal content.