Research projects

VC-dimension of cofinal families

This project arose from my Master's thesis, and concerns itself with the following question: what is the minimal VC-dimension of a family of sets of some bounded size, such that any set of some fixed other size is contained in one of its members? The first paper in this project deals with the finitary case and is purely combinatorial. The second paper deals with the infinitary case, and is completely concerned with the following order-theoretic construction: a well-order on a set, on each of its initial segments an additional well-order, on each of the initial segments of these smaller orders an additional well-order, etc. The paper explores various contexts in which one can construct such systems with certain desired properties, and both is motivated by and uses techniques from first-order logic (in particular, forcing). 

1. “Set Systems with Covering Properties and Low VC-dimension”, with Johanna Steinmeyer, Graphs and Combinatorics (2025). ArXiv / Journal

2. "Cofinal Families of Finite VC-Dimension", with Omer Ben-Neria and Itay Kaplan, preprint. ArXiv

Descriptive set theory of the orbit equivalence relations of locally compact group actions

This project deals with some applications of descriptive set theory to the extensions of theories in nonsingular ergodic theory from their previously understood contexts of countable group actions to locally compact group actions. The first paper deals with the Hopf decompositions, and extends the various known characterizations of dissipative systems to the locally compact context. The second paper develops a new, more streamlined viewpoint on the notion of a hyperfinite equivalence relation in the context of free locally compact group actions, and how it relates to the celebrated Connes-Feldman-Weiss and Ornstein-Weiss theorems.

1. "The Hopf Decomposition", with Nachi Avraham-Re'em, to appear in Journal d'Analyse Mathématique. ArXiv

2. "Uncountable Hyperfiniteness and the Random Ratio Ergodic Theorem", with Nachi Avraham-Re'em, preprint. ArXiv

Miscellaneous

1. "Irrationality of the Length Spectrum", with Guy Sapire, Comptes Rendus Mathématique (2025). ArXiv / Journal