More Like this

1. MONOID ACTIONS AND ULTRAFILTER METHODS IN RAMSEY THEORY

   https://doi.org/10.1017/fms.2018.28  

   SOLECKI, SŁAWOMIR (January 2019, Forum of Mathematics, Sigma)

   First, we prove a theorem on dynamics of actions of monoids by endomorphisms of semigroups. Second, we introduce algebraic structures suitable for formalizing infinitary Ramsey statements and prove a theorem that such statements are implied by the existence of appropriate homomorphisms between the algebraic structures. We make a connection between the two themes above, which allows us to prove some general Ramsey theorems for sequences. We give a new proof of the Furstenberg–Katznelson Ramsey theorem; in fact, we obtain a version of this theorem that is stronger than the original one. We answer in the negative a question of Lupini on possible extensions of Gowers’ Ramsey theorem. 

   more » « less
2. Katok’s special representation theorem for multidimensional Borel flows

   https://doi.org/10.1017/etds.2023.62  

   SLUTSKY, KONSTANTIN (October 2023, Ergodic Theory and Dynamical Systems)

   Katok’s special representation theorem states that any free ergodic measure-preserving R^d-flow can be realized as a special flow over a Z^d-action. It provides a multidimensional generalization of the ‘flow under a function’ construction. We prove the analog of Katok’s theorem in the framework of Borel dynamics and show that, likewise, all free Borel R^d-flows emerge from Z^d-actions through the special flow construction using bi-Lipschitz cocycles. 

   more » « less
3. Identifying 1-rectifiable measures in Carnot groups

   https://doi.org/10.1515/agms-2023-0102  

   Badger, Matthew; Li, Sean; Zimmerman, Scott (November 2023, Analysis and Geometry in Metric Spaces)

   We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of the first author and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in R^2 (P. Jones, 1990), in R^n (K. Okikolu, 1992), or in an arbitrary Carnot group (the second author) in terms of local geometric least squares data called Jones' beta numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in R^n that charges a rectifiable curve in an arbitrary complete, doubling, locally quasiconvex metric space. 

   more » « less
4. Borel asymptotic dimension and hyperfinite equivalence relations

   https://doi.org/10.1215/00127094-2022-0100  

   Conley, Clinton T; Jackson, Steve C; Marks, Andrew S; Seward, Brandon M; Tucker-Drob, Robin D (November 2023, Duke Mathematical Journal)

   A long-standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we prove that this question always has a positive answer when the acting group is polycyclic, and we obtain a positive answer for all free actions of a large class of solvable groups including the Baumslag–Solitar group BS(1, 2) and the lamplighter group Z2 ≀ Z. This marks the first time that a group of exponential volume-growth has been verified to have this property. In obtaining this result we introduce a new tool for studying Borel equivalence relations by extending Gromov’s notion of asymptotic dimension to the Borel setting. We show that countable Borel equivalence relations of finite Borel asymptotic dimension are hyperfinite, and more generally we prove under a mild compatibility assumption that increasing unions of such equivalence relations are hyperfinite. As part of our main theorem, we prove for a large class of solvable groups that all of their free Borel actions have finite Borel asymptotic dimension (and finite dynamic asymptotic dimension in the case of a continuous action on a zero dimensional space). We also provide applications to Borel chromatic numbers, Borel and continuous Følner tilings, topological dynamics, and C∗-algebras. 

   more » « less
5. Borel line graphs

   https://doi.org/10.1017/jsl.2024.50  

   ANDERSON, JAMES; BERNSHTEYN, ANTON (November 2024, The Journal of Symbolic Logic)

   Abstract We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the nine finite graphs from the classical result of Beineke together with a 10th infinite graph associated with the equivalence relation$$\mathbb {E}_0$$on the Cantor space. As a corollary, we prove a partial converse to the Feldman–Moore theorem, which allows us to characterize all locally countable Borel line graphs in terms of their Borel chromatic numbers. 

   more » « less