Bi-Amalgamation of Small Weak Global Dimension

arXiv (Cornell University), 2021

It is well known that a classical Fubini theorem for Hausdorff dimension cannot hold; that is, the dimension of the intersections of a fixed set with a parallel family of planes do not determine the dimension of the set. Here we prove that a Fubini theorem for Hausdorff dimension does hold modulo sets that are small on all Lipschitz graphs. We say that G ⊂ R k × R n is Γ k-null if for every Lipschitz function f : R k → R n the set {t ∈ R k : (t, f (t)) ∈ G} has measure zero. We show that for every Borel set E ⊂ R k × R n with dim (proj R k E) = k there is a Γ k-null subset G ⊂ E such that dim (E \ G) = k + ess-sup(dim Et) where ess-sup(dim Et) is the essential supremum of the Hausdorff dimension of the vertical sections {Et} t∈R k of E. In addition, we show that, provided that E is not Γ k-null, there is a Γ k-null subset G ⊂ E such that for F = E \ G, the Fubini-property holds, that is, dim (F) = k + ess-sup(dim Ft). We also obtain more general results by replacing R k by an Ahlfors-David regular set. Applications of our results include Fubini-type results for unions of affine subspaces, connection to the Kakeya conjecture and projection theorems. Contents 1. Introduction 1 2. Preliminaries and statements of our main results 2 2.1. Notation and basic definitions 2 2.2. The Fubini-type setting 3 2.3. Our main theorems 5 3. Applications to unions of affine subspaces and projection theorems 6 3.1. Results for unions of affine subspaces 6 3.2. Projection theorems as corollaries 9 4. Fubini property and counterexamples 5. Proofs of the main results 6. The proof of Lemma 5.4 7. The proof of Lemma 5.5 Acknowledgment Appendix A. The ZFC proof of Lemma 6.1 References

2013

In this note, we characterize the (weak) Gorenstein global dimension for arbitrary associative rings. Also, we extend the well-known Hilbert's syzygy Theorem to the weak Gorenstein global dimension, and we study the weak Goren-stein homological dimensions of direct product of rings which gives examples of non-coherent rings with finite Gorenstein dimensions > 0 and infinite classical weak dimension.

arXiv (Cornell University), 2022

The main goal of this note is to prove a coarse analogue of Factorization Theorems in Dimension Theory: Let f : X → Y be a coarsely continuous map. Then f factors through coarsely continuous maps g : X → Z and h : Z → Y with asymptotic dimension of Z at most asymptotic dimension of X and the weight of Z at most the weight of Y .

Illinois Journal of Mathematics

Motivated by connections between dimension groups and good measures or minimal actions on Cantor sets (especially Töplitz), we find realizations of classes of dimension groups as limits of primitive matrices all of which have equal column sums, or equal row sums.

2009

In this note we characterize the (resp., weak) Gorenstein global dimension for an arbitrary ring. Also, we extend the well-known Hilbert's syzygy Theorem to the weak Gorenstein global dimension and we study the weak Gorenstein homological dimensions of direct product of rings, which gives examples of non-coherent rings of finite Gorenstein dimensions $>0$ and infinite classical weak dimension.

Indiana University Mathematics Journal, 2020

We study the upper regularity dimension which describes the extremal local scaling behaviour of a measure and effectively quantifies the notion of doubling. We conduct a thorough study of the upper regularity dimension, including its relationship with other concepts such as the Assouad dimension, the upper local dimension, the L q -spectrum and weak tangent measures. We also compute the upper regularity dimension explicitly in a number of important contexts including self-similar measures, self-affine measures, and measures on sequences.

Topology and its Applications, 2004

We introduce splintered and strongly splintered spaces. They are generalizations of both almost zero-dimensional spaces and weakly 1-dimensional spaces. We prove that there are n-dimensional strongly splintered spaces for every n, and that there is a 1-dimensional splintered space X such that dim X n = n for every n. This solves a problem in the literature. Finally, we correct a flaw in an argument of Tomaszewski in his product formula for the dimension of the product of a weakly n-dimensional and a weakly m-dimensional space.

Journal of the Australian Mathematical Society, 2015

We prove some new bounds for the size of the maximal dissociated subset of structured (having small sumset, large energy and so on) subsets of an abelian group.

2005

Gromov \cite{Gr$_1$} and Dranishnikov \cite{Dr$_1$} introduced asymptotic and coarse dimensions of proper metric spaces via quite different ways. We define coarse and asymptotic dimension of all metric spaces in a unified manner and we investigate relationships between them generalizing results of Dranishnikov \cite{Dr$_1$} and Dranishnikov-Keesling-Uspienskij \cite{DKU}.

Proceedings of the American Mathematical Society, 1999

Oversteegen and Tymchatyn proved that homeomorphism groups of positive dimensional Menger compacta are 1 1 -dimensional by proving that almost 0 0 -dimensional spaces are at most 1 1 -dimensional. These homeomorphism groups are almost 0 0 -dimensional and at least 1 1 -dimensional by classical results of Brechner and Bestvina. In this note we prove that almost n n -dimensional spaces for n ≥ 1 n \geq 1 are n n -dimensional. As a corollary we answer in the affirmative an old question of R. Duda by proving that every hereditarily locally connected, non-degenerate, separable, metric space is 1 1 -dimensional.