Papers by Grzegorz Tomkowicz
Mathematische Annalen, Jul 20, 2023
Given a probability space (X , B, m), measure preserving transformations g 1 , . . . , g k of X ,... more Given a probability space (X , B, m), measure preserving transformations g 1 , . . . , g k of X , and a colour set C, a colouring rule is a way to colour the space with C such that the colours allowed for a point x are determined by that point's location in X and the colours of the finitely many g 1 (x), . . . , g k (x) (called descendants). We represent a colouring rule as a correspondence F defined on A colouring rule is paradoxical if it can be satisfied in some way almost everywhere with respect to m, but not in any way that is measurable with respect to a finitely additive measure that extends the probability measure m defined on B and for which the finitely many transformations g 1 , . . . , g k remain measure preserving. We show that a colouring rule can be paradoxical when the g 1 , . . . , g k are members of a semi-group G, the probability space X and the colour set C are compact sets, C is convex and finite dimensional, and the colouring rule says if c : X → C is the colouring function then the colour c(x) must lie (m a.e.) in F(x, c(g 1 (x)), . . . , c(g k (x))) for a non-empty upper-semi-continuous convex-valued correspondence F. Furthermore we show that this colouring rule has a stability property-there is a positive small enough so that if the expected deviation from the rule does not exceed then the colouring cannot be measurable in the same finitely additive way. As a consequence, there is a twoperson Bayesian game with equilibria, but all -equilibria for small enough are not Dedicated to
Locally Commutative Actions: Minimizing the Number of Pieces in a Paradoxical Decomposition
Cambridge University Press eBooks, Jun 5, 2016
Euclidean Transformation Groups
Cambridge University Press eBooks, Jun 5, 2016
Paradoxes in Low Dimensions
Cambridge University Press eBooks, Jun 5, 2016
Addendum to the Foreword
Cambridge University Press eBooks, Jun 5, 2016
Addendum to the Foreword
The Banach–Tarski Paradox
On bounded paradoxical sets and Lie groups
Geometriae dedicata, Apr 14, 2024
On some 2-edge-connected homogenous graphs and Cayley graphs with two removable vertices
Acta Mathematica Hungarica, Feb 6, 2021
A graph $$\Gamma$$ Γ is called homogenous if the group of automorphisms of $$\Gamma$$ Γ acts tran... more A graph $$\Gamma$$ Γ is called homogenous if the group of automorphisms of $$\Gamma$$ Γ acts transitively on the set of vertices of $$\Gamma$$ Γ . We will study the following properties for homogenous graphs: Removability : a vertex p is removable if the graph obtained by removing it is isomorphic to the original. Property (R): for every edge x there exists an edge y such that removing x and y disconnects the graph. We will study also Cayley graphs of groups of automorphisms of graphs with the property (R) and related concepts.
arXiv (Cornell University), Mar 21, 2022
Given a probability space (X, B, m), measure preserving transformations g 1 ,. .. , g k of X, and... more Given a probability space (X, B, m), measure preserving transformations g 1 ,. .. , g k of X, and a colour set C, a colouring rule is a way to colour the space with C such that the colours allowed for a point x are determined by that point's location and the colours of the finitely g 1 (x),. .. , g k (x) with g i (x) = x for all i and almost all x. We represent a colouring rule as a correspondence F defined on X × C k with values in C. A function f : X → C satisfies the rule at x if f (x) ∈ F (x, f (g 1 x),. .. , f (g k x)). A colouring rule is paradoxical if it can be satisfied in some way almost everywhere with respect to m, but not in any way that is measurable with respect to a finitely additive measure that extends the probability measure m defined on B and for which the finitely many transformations g 1 ,. .. , g k remain measure preserving. Can a colouring rule be paradoxical if both X and the colour set C are convex and compact sets and the colouring rule says if c : X → C is the colouring function then the colour c(x) must lie (m a.e.) in F (x, c(g 1 (x)),. .. , c(g k (x))) for a non-empty upper-semi-continuous convex-valued correspondence F defined on X × C k ? The answer is yes, and we present such an example. We show that this result is robust, including that any colouring that approximates the correspondence by ǫ for small enough positive ǫ also cannot be measurable in the same finitely additive way. Because non-empty upper-semi-continuous convex-valued correspondences on Euclidean space can be approximated by continuous functions, there are paradoxical colouring rules that are defined by continuous functions.
arXiv (Cornell University), May 14, 2018
We colour every point x of a probability space X according to the colours of a finite list x 1 , ... more We colour every point x of a probability space X according to the colours of a finite list x 1 , x 2 ,. .. , x k of points such that each of the x i , as a function of x, is a measure preserving transformation. We ask two questions about a colouring rule: (1) does there exist a finitely additive extension of the probability measure for which the x i remain measure preserving and also a colouring obeying the rule almost everywhere that is measurable with respect to this extension?, and (2) does there exist any colouring obeying the rule almost everywhere? If the answer to the first question is no and to the second question yes, we say that the colouring rule is paradoxical. A paradoxical colouring rule not only allows for a paradoxical partition of the space, it requires one. We pay special attention to generalizations of the Hausdorff paradox.
arXiv (Cornell University), Jan 27, 2017
We present a three player Bayesian game for which there is no ǫ-equilibria in Borel measurable st... more We present a three player Bayesian game for which there is no ǫ-equilibria in Borel measurable strategies for small enough positive ǫ, however there are non-measurable equilibria.
arXiv (Cornell University), May 28, 2021
Given a probability space (X, B, m), measure preserving transformations g 1 ,. .. , g k of X, and... more Given a probability space (X, B, m), measure preserving transformations g 1 ,. .. , g k of X, and a colour set C, a colouring rule is a way to colour the space with C such that the colours allowed for a point x are determined by that point's location and the colours of the finitely g 1 (x),. .. , g k (x) with g i (x) = x for all i and almost all x. We represent a colouring rule as a correspondence F defined on X × C k with values in C. A function f : X → C satisfies the rule at x if f (x) ∈ F (x, f (g 1 x),. .. , f (g k x)). A colouring rule is paradoxical if it can be satisfied in some way almost everywhere with respect to m, but not in any way that is measurable with respect to a finitely additive measure that extends the probability measure m and for which the finitely many transformations g 1 ,. .. , g k remain measure preserving. We show that a colouring rule can be paradoxical when the g 1 ,. .. , g k are members of a group G, the probability space X and the colour set C are compact sets, C is convex and finite dimensional, and the colouring rule says if c : X → C is the colouring function then the colour c(x) must lie (m a.e.) in F (x, c(g 1 (x)),. .. , c(g k (x))) for a non-empty upper-semi-continuous convex-valued correspondence F defined on X × C k. We show that any colouring that approximates the correspondence by ǫ for small enough positive ǫ cannot be measurable in the same finitely additive way. Furthermore any function satisfying the colouring rule illustrates a paradox through finitely many measure preserving shifts defining injective maps from the whole space to subsets of measure summing up to less than one.
A Continuous Paradoxical Colouring Rule Using Group Action
arXiv (Cornell University), May 28, 2021
arXiv (Cornell University), Nov 9, 2022
We colour every point x of a probability space X according to the colours of a finite list x 1 , ... more We colour every point x of a probability space X according to the colours of a finite list x 1 , x 2 ,. .. , x k of points such that each of the x i , as a function of x, is a measure preserving transformation. We ask two questions about a colouring rule: (1) does there exist a finitely additive extension of the probability measure for which the x i remain measure preserving and also a colouring obeying the rule almost everywhere that is measurable with respect to this extension?, and (2) does there exist some colouring obeying the rule almost everywhere? If the answer to the first question is no and to the second question yes, we say that the colouring rule is paradoxical. A paradoxical colouring rule not only allows for a paradoxical partition of the space, it requires one. If a colouring rule is paradoxical, an axiom of choice is used to prove (2) but not typically used to prove (1). We show that a form of paradoxical decomposition can be created from the colour classes of any such colouring. We show that proper vertex colouring can be paradoxical. We conclude with several new topics and open questions.
Geometriae Dedicata, 2018
We apply a construction of G. A. Margulis to show that there exists a free nonabelian properly di... more We apply a construction of G. A. Margulis to show that there exists a free nonabelian properly discontinuous group of affine transformations of R 3 with both linear and translational parts having integer entries and acting on R 3 without fixed points.
Free Groups of Large Rank: Getting a Continuum of Spheres from One
The Banach–Tarski Paradox
Transition
The Banach–Tarski Paradox
The Semigroup of Equidecomposability Types
The Banach–Tarski Paradox
Applications of Amenability
The Banach–Tarski Paradox
Higher Dimensions
The Banach–Tarski Paradox
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Papers by Grzegorz Tomkowicz