Pub Date : 2019-12-31DOI: 10.1515/9780691188997-024
{"title":"Chapter 22. Existence of Many Large Orbits","authors":"","doi":"10.1515/9780691188997-024","DOIUrl":"https://doi.org/10.1515/9780691188997-024","url":null,"abstract":"","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126072652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-31DOI: 10.1515/9780691188997-023
{"title":"Chapter 21. Existence of Infinite Orbits","authors":"","doi":"10.1515/9780691188997-023","DOIUrl":"https://doi.org/10.1515/9780691188997-023","url":null,"abstract":"","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134554439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-31DOI: 10.1515/9780691188997-026
{"title":"Chapter 24. Some Elementary Number Theory","authors":"","doi":"10.1515/9780691188997-026","DOIUrl":"https://doi.org/10.1515/9780691188997-026","url":null,"abstract":"","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114392099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-02-19DOI: 10.23943/princeton/9780691181387.003.0027
R. Schwartz
This chapter proves the core case of the Copy Lemma. The proof follows the same strategy as in Chapter 25, but it is considerably harder. Section 26.2 proves the first two statements of the Copy Lemma. The rest of the chapter is devoted to the third statement. Section 26.3 defines geometric and arithmetic alignment as in the previous chapter, but with the twist that there are some extra indices that have to be looked after carefully. Section 26.4 verifies the geometric alignment criterion just as in the previous chapter. Section 26.5 shows that the signs of the two capacity sequences match. Section 26.6 presents a technical lemma to deal with the mass sequences. Section 26.7 shows that the signs of the mass sequences agree on the central indices. Section 26.8 verifies that the signs of the mass sequences agree on the peripheral sequences except for two special indices where the signs can disagree. Section 26.9 finishes the proof of this long and difficult argument.
{"title":"The Core Case","authors":"R. Schwartz","doi":"10.23943/princeton/9780691181387.003.0027","DOIUrl":"https://doi.org/10.23943/princeton/9780691181387.003.0027","url":null,"abstract":"This chapter proves the core case of the Copy Lemma. The proof follows the same strategy as in Chapter 25, but it is considerably harder. Section 26.2 proves the first two statements of the Copy Lemma. The rest of the chapter is devoted to the third statement. Section 26.3 defines geometric and arithmetic alignment as in the previous chapter, but with the twist that there are some extra indices that have to be looked after carefully. Section 26.4 verifies the geometric alignment criterion just as in the previous chapter. Section 26.5 shows that the signs of the two capacity sequences match. Section 26.6 presents a technical lemma to deal with the mass sequences. Section 26.7 shows that the signs of the mass sequences agree on the central indices. Section 26.8 verifies that the signs of the mass sequences agree on the peripheral sequences except for two special indices where the signs can disagree. Section 26.9 finishes the proof of this long and difficult argument.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124578211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-02-19DOI: 10.23943/princeton/9780691181387.003.0003
R. Schwartz
This chapter derives some basic properties of the plaid model. It is organized as follows. Section 2.2 deals with the symmetries of the plaid model. First, it deals with the unoriented model and then considers the oriented model. Section 2.3 proves the technical lemma that each unit integer segment contains exactly two intersection points. The work in this section reveals the nice geometric way that the slanting lines intersect each unit integer square. Section 2.4 establishes the following result: Within each block, there are exactly two lines of capacity k for each even k ɛ [0, ω]. Moreover, within the block, each line of capacity k has exactly k light points on it (when double points are appropriately counted). Section 2.5 establishes a subtle additional symmetry of the plaid model.
{"title":"Properties of the Model","authors":"R. Schwartz","doi":"10.23943/princeton/9780691181387.003.0003","DOIUrl":"https://doi.org/10.23943/princeton/9780691181387.003.0003","url":null,"abstract":"This chapter derives some basic properties of the plaid model. It is organized as follows. Section 2.2 deals with the symmetries of the plaid model. First, it deals with the unoriented model and then considers the oriented model. Section 2.3 proves the technical lemma that each unit integer segment contains exactly two intersection points. The work in this section reveals the nice geometric way that the slanting lines intersect each unit integer square. Section 2.4 establishes the following result: Within each block, there are exactly two lines of capacity k for each even k ɛ [0, ω]. Moreover, within the block, each line of capacity k has exactly k light points on it (when double points are appropriately counted). Section 2.5 establishes a subtle additional symmetry of the plaid model.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127894121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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