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Tomasz Cieśla

Publications and source records attributed to Tomasz Cieśla.

5 recordsLinked to original sources
arXiv2021

Measurable Hall's theorem for actions of abelian groups

We prove a measurable version of the Hall marriage theorem for actions of finitely generated abelian groups. In particular, it implies that for free measure-preserving actions of such groups, if two equidistributed measurable sets are equidecomposable, then they are equidecomposable using measurable pieces. The latter generalizes a recent result of Grabowski, Máthé and Pikhurko on the measurable circle squaring and confirms a special case of a conjecture of Gardner.

math.LO↗
arXiv2021

On random compact sets, equidecomposition, and domains of expansion in R^3

We study random compact subsets of R^3 which can be described as "random Menger sponges". We use those random sets to construct a pair of compact sets A and B in R^3 which are of the same positive measure, such that A can be covered by finitely many translates of B, B can be covered by finitely many translates of A, and yet A and B are not equidecomposable. Furthermore, we construct the first example of a compact subset of R^3 of positive measure which is not a domain of expansion. This answers a question of Adrian Ioana.

math.MG↗
arXiv2020

On lifting invariant probability measures

In this note we study when an invariant probability measure lifts to an invariant measure. Consider a standard Borel space $X$, a Borel probability measure $μ$ on $X$, a Borel map $T \colon X \to X$ preserving $μ$, a compact metric space $Y$, a continuous map $S\colon Y \to Y$, and a Borel surjection $p \colon Y \to X$ with $p\circ S = T \circ p$. We prove that if fibers of $p$ are compact then $μ$ lifts to an $S$-invariant measure on $Y$.

math.DS↗
arXiv2017

A filter on a collection of finite sets and Eberlein compacta

We introduce a $σ$-ideal on $ω_1 \times ω_1$ and a filter on the collection of graphs of strictly decreasing partial functions on $ω_1$ taking values in $ω_1$. We use them to prove that a certain space is a non-bisequential Eberlein compactum.

math.GN↗