302 research outputs found
Baire measurable paradoxical decompositions via matchings
We show that every locally finite bipartite Borel graph satisfying a
strengthening of Hall's condition has a Borel perfect matching on some comeager
invariant Borel set. We apply this to show that if a group acting by Borel
automorphisms on a Polish space has a paradoxical decomposition, then it admits
a paradoxical decomposition using pieces having the Baire property. This
strengthens a theorem of Dougherty and Foreman who showed that there is a
paradoxical decomposition of the unit ball in using Baire
measurable pieces. We also obtain a Baire category solution to the dynamical
von Neumann-Day problem: if is a nonamenable action of a group on a Polish
space by Borel automorphisms, then there is a free Baire measurable action
of on which is Lipschitz with respect to .Comment: Minor revision
Borel circle squaring
We give a completely constructive solution to Tarski's circle squaring
problem. More generally, we prove a Borel version of an equidecomposition
theorem due to Laczkovich. If and are
bounded Borel sets with the same positive Lebesgue measure whose boundaries
have upper Minkowski dimension less than , then and are
equidecomposable by translations using Borel pieces. This answers a question of
Wagon. Our proof uses ideas from the study of flows in graphs, and a recent
result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the
hyperfiniteness of free Borel actions of .Comment: Minor typos correcte
Measurable realizations of abstract systems of congruences
An abstract system of congruences describes a way of partitioning a space
into finitely many pieces satisfying certain congruence relations. Examples of
abstract systems of congruences include paradoxical decompositions and
-divisibility of actions. We consider the general question of when there are
realizations of abstract systems of congruences satisfying various
measurability constraints. We completely characterize which abstract systems of
congruences can be realized by nonmeager Baire measurable pieces of the sphere
under the action of rotations on the -sphere. This answers a question of
Wagon. We also construct Borel realizations of abstract systems of congruences
for the action of on .
The combinatorial underpinnings of our proof are certain types of decomposition
of Borel graphs into paths. We also use these decompositions to obtain some
results about measurable unfriendly colorings.Comment: minor correction
Stationary Reflection and the failure of SCH
In this paper we prove that from large cardinals it is consistent that there
is a singular strong limit cardinal such that the singular cardinal
hypothesis fails at and every collection of fewer than
stationary subsets of reflects simultaneously. For uncountable
cofinality, this situation was not previously known to be consistent. Using
different methods, we reduce the upper bound on the consistency strength of
this situation for to below a single partially
supercompact cardinal. The previous upper bound of infinitely many supercompact
cardinals was due to Sharon.Comment: 23 page
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