{"title":"A dynamical approach to nonhomogeneous spectra","authors":"Jun Yu Li, Xianjuan Liang","doi":"10.4064/fm191-5-2023","DOIUrl":"https://doi.org/10.4064/fm191-5-2023","url":null,"abstract":"Let $alpha>0$ and $0<gamma<1$. Define $g_{alpha,gamma}colon mathbb{N}tomathbb{N}_0$ by $g_{alpha,gamma}(n)=lfloor nalpha +gammarfloor$, where $lfloor x rfloor$ is the largest integer less than or equal to $x$. The set $g_{alpha,gamma}(mathbb{N})={g_{alpha,gamma}(n)colon ninmathbb{N}}$ is called the $gamma$-nonhomogeneous spectrum of $alpha$. By extension, the functions $g_{alpha,gamma}$ are referred to as spectra. In 1996, Bergelson, Hindman and Kra showed that the functions $g_{alpha,gamma}$ preserve some largeness of subsets of $mathbb{N}$, that is, if a subset $A$ of $mathbb{N}$ is an IP-set, a central set, an IP$^*$-set, or a central$^*$-set, then $g_{alpha,gamma}(A)$ is the corresponding object for all $alpha>0$ and $0<gamma<1$. In 2012, Hindman and Johnson extended this result to include several other notions of largeness: C-sets, J-sets, strongly central sets, and piecewise syndetic sets. We adopt a dynamical approach to this issue and build a correspondence between the preservation of spectra and the lift property of suspension. As an application, we give a unified proof of some known results and also obtain some new results.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46645559","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}
We first identify (up to linear isomorphism) the Lipschitz free spaces of quasiarcs. By decomposing quasiconformal trees into quasiarcs as done in an article of David, Eriksson-Bique, and Vellis, we then identify the Lipschitz free spaces of quasiconformal trees and prove that quasiconformal trees have Lipschitz dimension 1. Generalizing the aforementioned decomposition, we define a geometric tree-like decomposition of a metric space. Our results pertaining to quasiconformal trees are in fact special cases of results about metric spaces admitting a geometric tree-like decomposition. Furthermore, the methods employed in our study of Lipschitz free spaces yield a decomposition of any (weak) quasiarc into rectifiable and purely unrectifiable subsets, which may be of independent interest.
{"title":"Lipschitz functions on quasiconformal trees","authors":"D. Freeman, C. Gartland","doi":"10.4064/fm273-3-2023","DOIUrl":"https://doi.org/10.4064/fm273-3-2023","url":null,"abstract":"We first identify (up to linear isomorphism) the Lipschitz free spaces of quasiarcs. By decomposing quasiconformal trees into quasiarcs as done in an article of David, Eriksson-Bique, and Vellis, we then identify the Lipschitz free spaces of quasiconformal trees and prove that quasiconformal trees have Lipschitz dimension 1. Generalizing the aforementioned decomposition, we define a geometric tree-like decomposition of a metric space. Our results pertaining to quasiconformal trees are in fact special cases of results about metric spaces admitting a geometric tree-like decomposition. Furthermore, the methods employed in our study of Lipschitz free spaces yield a decomposition of any (weak) quasiarc into rectifiable and purely unrectifiable subsets, which may be of independent interest.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47344908","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}
Define the special tree number, denoted $mathfrak{st}$, to be the least size of a tree of height $omega_1$ which is neither special nor has a cofinal branch. This cardinal had previously been studied in the context of fragments of $mathsf{MA}$ but in this paper we look at its relation to other, more typical, cardinal characteristics. Classical facts imply that $aleph_1 leq mathfrak{st} leq 2^{aleph_0}$, under Martin's Axiom $mathfrak{st} = 2^{aleph_0}$ and that $mathfrak{st} = aleph_1$ is consistent with $mathsf{MA}({rm Knaster}) + 2^{aleph_0} = kappa$ for any regular $kappa$ thus the value of $mathfrak{st}$ is not decided by $mathsf{ZFC}$ and in fact can be strictly below essentially all well studied cardinal characteristics. We show that conversely it is consistent that $mathfrak{st} = 2^{aleph_0} = kappa$ for any $kappa$ of uncountable cofinality while ${rm non}(mathcal M) = mathfrak{a} = mathfrak{s} = mathfrak{g} = aleph_1$. In particular $mathfrak{st}$ is independent of the lefthand side of Cicho'{n}'s diagram, amongst other things. The proof involves an in depth study of the standard ccc forcing notion to specialize (wide) Aronszajn trees, which may be of independent interest.
{"title":"The special tree number","authors":"Corey Bacal Switzer","doi":"10.4064/fm180-1-2023","DOIUrl":"https://doi.org/10.4064/fm180-1-2023","url":null,"abstract":"Define the special tree number, denoted $mathfrak{st}$, to be the least size of a tree of height $omega_1$ which is neither special nor has a cofinal branch. This cardinal had previously been studied in the context of fragments of $mathsf{MA}$ but in this paper we look at its relation to other, more typical, cardinal characteristics. Classical facts imply that $aleph_1 leq mathfrak{st} leq 2^{aleph_0}$, under Martin's Axiom $mathfrak{st} = 2^{aleph_0}$ and that $mathfrak{st} = aleph_1$ is consistent with $mathsf{MA}({rm Knaster}) + 2^{aleph_0} = kappa$ for any regular $kappa$ thus the value of $mathfrak{st}$ is not decided by $mathsf{ZFC}$ and in fact can be strictly below essentially all well studied cardinal characteristics. We show that conversely it is consistent that $mathfrak{st} = 2^{aleph_0} = kappa$ for any $kappa$ of uncountable cofinality while ${rm non}(mathcal M) = mathfrak{a} = mathfrak{s} = mathfrak{g} = aleph_1$. In particular $mathfrak{st}$ is independent of the lefthand side of Cicho'{n}'s diagram, amongst other things. The proof involves an in depth study of the standard ccc forcing notion to specialize (wide) Aronszajn trees, which may be of independent interest.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49402358","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}
A. Carderi, Matthieu Joseph, F. L. Maitre, R. Tessera
Belinskaya's theorem states that given an ergodic measure-preserving transformation, any other transformation with the same orbits and an $mathrm{L}^1$ cocycle must be flip-conjugate to it. Our main result shows that this theorem is optimal: for all $p<1$ the integrability condition on the cocycle cannot be relaxed to being in $mathrm{L}^p$. This also allows us to answer a question of Kerr and Li: for ergodic measure-preserving transformations, Shannon orbit equivalence doesn't boil down to flip-conjugacy.
{"title":"Belinskaya’s theorem is optimal","authors":"A. Carderi, Matthieu Joseph, F. L. Maitre, R. Tessera","doi":"10.4064/fm266-4-2023","DOIUrl":"https://doi.org/10.4064/fm266-4-2023","url":null,"abstract":"Belinskaya's theorem states that given an ergodic measure-preserving transformation, any other transformation with the same orbits and an $mathrm{L}^1$ cocycle must be flip-conjugate to it. Our main result shows that this theorem is optimal: for all $p<1$ the integrability condition on the cocycle cannot be relaxed to being in $mathrm{L}^p$. This also allows us to answer a question of Kerr and Li: for ergodic measure-preserving transformations, Shannon orbit equivalence doesn't boil down to flip-conjugacy.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48728765","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}
Let $K$ be a type-definable infinite field in an NIP theory. If $K$ has characteristic $p>0$, then $K$ is Artin-Schreier closed (it has no Artin-Schreier extensions). As a consequence, $p$ does not divide the degree of any finite separable extension of $K$. This generalizes a theorem of Kaplan, Scanlon, and Wagner.
{"title":"Type-definable NIP fields are Artin–Schreier closed","authors":"Will Johnson","doi":"10.4064/fm149-8-2022","DOIUrl":"https://doi.org/10.4064/fm149-8-2022","url":null,"abstract":"Let $K$ be a type-definable infinite field in an NIP theory. If $K$ has characteristic $p>0$, then $K$ is Artin-Schreier closed (it has no Artin-Schreier extensions). As a consequence, $p$ does not divide the degree of any finite separable extension of $K$. This generalizes a theorem of Kaplan, Scanlon, and Wagner.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47844518","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}
{"title":"On a conjecture of Debs and Saint Raymond","authors":"A. Kwela","doi":"10.4064/fm111-5-2022","DOIUrl":"https://doi.org/10.4064/fm111-5-2022","url":null,"abstract":"","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70171624","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}
{"title":"Special groups and quadratic forms over rings with non-zero-divisor coefficients","authors":"M. Dickmann, F. Miraglia, H. Ribeiro","doi":"10.4064/fm137-12-2021","DOIUrl":"https://doi.org/10.4064/fm137-12-2021","url":null,"abstract":"","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70194424","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}
{"title":"Increasing sequences of principal left ideals of $beta mathbb{Z}$ are finite","authors":"Y. Zelenyuk","doi":"10.4064/fm17-8-2021","DOIUrl":"https://doi.org/10.4064/fm17-8-2021","url":null,"abstract":"","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224110","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}
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