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Kamran Reihani - One of the best experts on this subject based on the ideXlab platform.
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$K$-theory of Furstenberg Transformation Group $C^*$-algebras
Canadian Journal of Mathematics, 2013Co-Authors: Kamran ReihaniAbstract:AbstractThis paper studies the K-theoretic invariants of the crossed product C*-algebras associated with an important family of homeomorphisms of the tori Tn called Furstenberg transformations. Using the Pimsner–Voiculescu theorem, we prove that given n, the K-groups of those crossed products whose corresponding n × n integer matrices are unipotent of maximal degree always have the same rank an. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these K-groups is false. Using the representation theory of the simple Lie algebra sl(2;C), we show that, remarkably, an has a combinatorial significance. For example, every a2n+1 is just the number of ways that 0 can be represented as a sum of integers between–n and n (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdős), a simple explicit formula for the asymptotic behavior of the sequence {an} is given. Finally, we describe the order structure of the K0-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.
A. Jabbari - One of the best experts on this subject based on the ideXlab platform.
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On Lau–Loy’s decomposition of a measure algebra on CHART groups
Periodica Mathematica Hungarica, 2020Co-Authors: Z. Bahramian, A. JabbariAbstract:By using the Furstenberg–Ellis–Namioka structure theorem, we give a decomposition theorem for the Banach algebra $${\mathcal {M}}(G)$$ M ( G ) , i.e. the Banach algebra of those complex regular Borel measures on a compact Hausdorff admissible right topological (or simply CHART) group G for which the natural convolution product makes sense, generalizing an existing result due to Lau and Loy. Next, we characterize the Furstenberg–Ellis–Namioka structure theorem on a family of CHART groups, namely the groups $$E({\mathbb {T}})^{k}$$ E ( T ) k where $$E({\mathbb {T}})$$ E ( T ) is the family of all endomorphisms of the unit circle $${\mathbb {T}}$$ T , and then we apply the generalized decomposition theorem to these groups.
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Furstenberg–Ellis–Namioka Structure Theorem on a CHART Group
Bulletin of the Iranian Mathematical Society, 2018Co-Authors: M. Zaman-abadi, A. JabbariAbstract:For $$E({\mathbb {T}})$$ being the endomorphism group of the circle group $${\mathbb {T}}$$ , the Furstenberg–Ellis–Namioka Structure Theorem of the CHART group $$G=E({\mathbb {T}})\times {\mathbb {T}}$$ with the product $$(f,u)(g,v)=(fg,uvf\circ g(\mathrm{e}^{i}))$$ is known to be equal to $$\{G,1_{\mathbb {T}}\times {\mathbb {T}},\{(1_{\mathbb {T}},1)\}\}$$ . A somewhat similar group structure is known to exist on $$E({\mathbb {T}})\times E({\mathbb {T}})\times {\mathbb {T}}$$ , studied by Milnes. We give an explicit characterization of the Furstenberg–Ellis–Namioka Structure Theorem for an admissible subgroup $$\Sigma $$ of $$E({\mathbb {T}})\times E({\mathbb {T}})\times {\mathbb {T}}$$ , where $$\Sigma $$ is the Ellis group of the Hahn-type skew product dynamical system on the 3-torus $${\mathbb {T}}^3$$ .
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Furstenberg ellis namioka structure theorem on a chart group
Bulletin of The Iranian Mathematical Society, 2018Co-Authors: M Zamanabadi, A. JabbariAbstract:For $$E({\mathbb {T}})$$ being the endomorphism group of the circle group $${\mathbb {T}}$$ , the Furstenberg–Ellis–Namioka Structure Theorem of the CHART group $$G=E({\mathbb {T}})\times {\mathbb {T}}$$ with the product $$(f,u)(g,v)=(fg,uvf\circ g(\mathrm{e}^{i}))$$ is known to be equal to $$\{G,1_{\mathbb {T}}\times {\mathbb {T}},\{(1_{\mathbb {T}},1)\}\}$$ . A somewhat similar group structure is known to exist on $$E({\mathbb {T}})\times E({\mathbb {T}})\times {\mathbb {T}}$$ , studied by Milnes. We give an explicit characterization of the Furstenberg–Ellis–Namioka Structure Theorem for an admissible subgroup $$\Sigma $$ of $$E({\mathbb {T}})\times E({\mathbb {T}})\times {\mathbb {T}}$$ , where $$\Sigma $$ is the Ellis group of the Hahn-type skew product dynamical system on the 3-torus $${\mathbb {T}}^3$$ .
Ezequiel Rela - One of the best experts on this subject based on the ideXlab platform.
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Small Furstenberg sets
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Ursula Molter, Ezequiel RelaAbstract:Abstract For α in ( 0 , 1 ] , a subset E of R 2 is called a Furstenberg set of type α or F α -set if for each direction e in the unit circle there is a line segment l e in the direction of e such that the Hausdorff dimension of the set E ∩ l e is greater than or equal to α . In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of zero -dimensional Furstenberg type sets. Namely, for h γ ( x ) = log − γ ( 1 x ) , γ > 0 , we construct a set E γ ∈ F h γ of Hausdorff dimension not greater than 1 2 . Since in a previous work we showed that 1 2 is a lower bound for the Hausdorff dimension of any E ∈ F h γ , with the present construction, the value 1 2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions h γ .
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Furstenberg sets for a fractal set of directions
arXiv: Classical Analysis and ODEs, 2010Co-Authors: Ursula Molter, Ezequiel RelaAbstract:In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $\alpha,\beta\in(0,1]$, we will say that a set $E\subset \R^2$ is an $F_{\alpha\beta}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $\beta$ and, for each direction $e$ in $L$, there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is equal or greater than $\alpha$. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that $\dim(E)\ge\max\left\{\alpha+\frac{\beta}{2} ; 2\alpha+\beta -1\right\}$ for any $E\in F_{\alpha\beta}$. In particular we are able to extend previously known results to the ``endpoint'' $\alpha=0$ case.
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Small Furstenberg sets
arXiv: Classical Analysis and ODEs, 2010Co-Authors: Ursula Molter, Ezequiel RelaAbstract:For $\alpha$ in $(0,1]$, a subset $E$ of $\RR$ is called Furstenberg set of type $\alpha$ or $F_\alpha$-set if for each direction $e$ in the unit circle there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is greater or equal than $\alpha$. In this paper we show that if $\alpha > 0$, there exists a set $E\in F_\alpha$ such that $\HH{g}(E)=0$ for $g(x)=x^{1/2+3/2\alpha}\log^{-\theta}(\frac{1}{x})$, $\theta>\frac{1+3\alpha}{2}$, which improves on the the previously known bound, that $H^{\beta}(E) = 0$ for $\beta>1/2+3/2\alpha$. Further, by refining the argument in a subtle way, we are able to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for $\h_\gamma(x)=\log^{-\gamma}(\frac{1}{x})$, $\gamma>0$, we construct a set $E_\gamma\in F_{\h_\gamma}$ of Hausdorff dimension not greater than 1/2. Since in a previous work we showed that 1/2 is a lower bound for the Hausdorff dimension of any $E\in F_{\h_\gamma}$, with the present construction, the value 1/2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions $\h_\gamma$.
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Improving dimension estimates for Furstenberg-type sets
Advances in Mathematics, 2010Co-Authors: Ursula Molter, Ezequiel RelaAbstract:In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α ∈ (0, 1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment le in the direction of e for which dimH(le ∩ F ) ≥ α. It is well known that dimH(F ) ≥ max{2α, α + 1 } - and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures H h defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets. The main difficulty we had to overcome, was that if H h (F ) = 0, there always exists g ≺ h such that H g (F ) = 0 (here ≺ refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true step down from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint α = 0.
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improving dimension estimates for Furstenberg type sets
arXiv: Classical Analysis and ODEs, 2009Co-Authors: Ursula Molter, Ezequiel RelaAbstract:In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional growth conditions on the dimension function, we obtain a lower bound on the dimension of "zero dimensional" Furstenberg sets.
W T Gowers - One of the best experts on this subject based on the ideXlab platform.
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polymath and the density hales jewett theorem
2010Co-Authors: W T GowersAbstract:Van der Waerden’s theorem has two well-known and very different generalizations. One is the Hales-Jewett theorem, one of the cornerstones of Ramsey theory. The other is Endre Szemeredi’s famous density version of the theorem, which has played a pivotal role in the recent growth of additive combinatorics. In 1991 Furstenberg and Katznelson proved the density Hales-Jewett theorem, a result that has the same relationship to the Hales-Jewett theorem that Szemeredi’s theorem has to van der Waerden’s theorem. Furstenberg and Katznelson used a development of the ergodic-theoretic machinery introduced by Furstenberg. Very recently, a new and much more elementary proof was discovered in a rather unusual way - by a collaborative process carried out in the open with the help of blogs and a wiki. In this informal paper, we briefly discuss this discovery process and then give a detailed sketch of the new proof.
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hypergraph regularity and the multidimensional szemeredi theorem
Annals of Mathematics, 2007Co-Authors: W T GowersAbstract:We prove analogues for hypergraphs of Szemeredi's regularity lemma and the associated counting lemma for graphs. As an application, we give the first combinatorial proof of the multidimensional Szemeredi theorem of Furstenberg and Katznelson, and the first proof that provides an explicit bound. Similar results with the same consequences have been obtained independently by Nagle, Rodl, Schacht and Skokan.
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hypergraph regularity and the multidimensional szemer edi theorem
arXiv: Combinatorics, 2007Co-Authors: W T GowersAbstract:We prove analogues for hypergraphs of Szemer\'edi's regularity lemma and the associated counting lemma for graphs. As an application, we give the first combinatorial proof of the multidimensional Szemer\'edi theorem of Furstenberg and Katznelson, and the first proof that provides an explicit bound. Similar results with the same consequences have been obtained independently by Nagle, R\"odl, Schacht and Skokan.
Ursula Molter - One of the best experts on this subject based on the ideXlab platform.
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Small Furstenberg sets
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Ursula Molter, Ezequiel RelaAbstract:Abstract For α in ( 0 , 1 ] , a subset E of R 2 is called a Furstenberg set of type α or F α -set if for each direction e in the unit circle there is a line segment l e in the direction of e such that the Hausdorff dimension of the set E ∩ l e is greater than or equal to α . In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of zero -dimensional Furstenberg type sets. Namely, for h γ ( x ) = log − γ ( 1 x ) , γ > 0 , we construct a set E γ ∈ F h γ of Hausdorff dimension not greater than 1 2 . Since in a previous work we showed that 1 2 is a lower bound for the Hausdorff dimension of any E ∈ F h γ , with the present construction, the value 1 2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions h γ .
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Furstenberg sets for a fractal set of directions
arXiv: Classical Analysis and ODEs, 2010Co-Authors: Ursula Molter, Ezequiel RelaAbstract:In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $\alpha,\beta\in(0,1]$, we will say that a set $E\subset \R^2$ is an $F_{\alpha\beta}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $\beta$ and, for each direction $e$ in $L$, there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is equal or greater than $\alpha$. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that $\dim(E)\ge\max\left\{\alpha+\frac{\beta}{2} ; 2\alpha+\beta -1\right\}$ for any $E\in F_{\alpha\beta}$. In particular we are able to extend previously known results to the ``endpoint'' $\alpha=0$ case.
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Small Furstenberg sets
arXiv: Classical Analysis and ODEs, 2010Co-Authors: Ursula Molter, Ezequiel RelaAbstract:For $\alpha$ in $(0,1]$, a subset $E$ of $\RR$ is called Furstenberg set of type $\alpha$ or $F_\alpha$-set if for each direction $e$ in the unit circle there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is greater or equal than $\alpha$. In this paper we show that if $\alpha > 0$, there exists a set $E\in F_\alpha$ such that $\HH{g}(E)=0$ for $g(x)=x^{1/2+3/2\alpha}\log^{-\theta}(\frac{1}{x})$, $\theta>\frac{1+3\alpha}{2}$, which improves on the the previously known bound, that $H^{\beta}(E) = 0$ for $\beta>1/2+3/2\alpha$. Further, by refining the argument in a subtle way, we are able to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for $\h_\gamma(x)=\log^{-\gamma}(\frac{1}{x})$, $\gamma>0$, we construct a set $E_\gamma\in F_{\h_\gamma}$ of Hausdorff dimension not greater than 1/2. Since in a previous work we showed that 1/2 is a lower bound for the Hausdorff dimension of any $E\in F_{\h_\gamma}$, with the present construction, the value 1/2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions $\h_\gamma$.
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Improving dimension estimates for Furstenberg-type sets
Advances in Mathematics, 2010Co-Authors: Ursula Molter, Ezequiel RelaAbstract:In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α ∈ (0, 1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment le in the direction of e for which dimH(le ∩ F ) ≥ α. It is well known that dimH(F ) ≥ max{2α, α + 1 } - and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures H h defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets. The main difficulty we had to overcome, was that if H h (F ) = 0, there always exists g ≺ h such that H g (F ) = 0 (here ≺ refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true step down from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint α = 0.
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improving dimension estimates for Furstenberg type sets
arXiv: Classical Analysis and ODEs, 2009Co-Authors: Ursula Molter, Ezequiel RelaAbstract:In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional growth conditions on the dimension function, we obtain a lower bound on the dimension of "zero dimensional" Furstenberg sets.
