Open sets satisfying systems of congruences

A famous result of Hausdorff states that a sphere with countably many points removed can be partitioned into three pieces A,B,C such that A is congruent to B (i.e., there is an isometry of the sphere which sends A to B), B is congruent to C, and A is congruent to (B union C); this result was the precursor of the Banach-Tarski paradox. Later, R. Robinson characterized the systems of congruences like this which could be realized by partitions of the (entire) sphere with rotations witnessing the congruences. The pieces involved were nonmeasurable. In the present paper, we consider the problem of which systems of congruences can be satisfied using open subsets of the sphere (or related spaces); of course, these open sets cannot form a partition of the sphere, but they can be required to cover "most of" the sphere in the sense that their union is dense. Various versions of the problem arise, depending on whether one uses all isometries of the sphere or restricts oneself to a free group of rotations (the latter version generalizes to many other suitable spaces), or whether one omits the requirement that the open sets have dense union, and so on. While some cases of these problems are solved by simple geometrical dissections, others involve complicated iterative constructions and/or results from the theory of free groups. Many interesting questions remain open.Comment: 44 page

Translating the Cantor set by a random

We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set "cancels randomness" in the sense that some of its members, when added to Martin-Lof random reals, identify a point with lower constructive dimension than the random itself. In particular, we find the Hausdorff dimension of the set of points in a Cantor set translate with a given constructive dimension

Narrow coverings of omega-product spaces

Results of Sierpinski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is "narrow" in a corresponding direction; that is, each line in that direction intersects the subset in a small set. For example, if the set (omega \times omega) is partitioned into two pieces along the diagonal, then one piece meets every horizontal line in a finite set, and the other piece meets each vertical line in a finite set. Such partitions or coverings can exist only when the sets forming the product are of limited size. This paper considers such coverings for products of infinitely many sets (usually a product of omega copies of the same cardinal kappa). In this case, a covering of the product by narrow sets, one for each coordinate direction, will exist no matter how large the factor sets are. But if one restricts the sets used in the covering (for instance, requiring them to be Borel in a product topology), then the existence of narrow coverings is related to a number of large cardinal properties: partition cardinals, the free subset problem, nonregular ultrafilters, and so on. One result given here is a relative consistency proof for a hypothesis used by S. Mrowka to construct a counterexample in the dimension theory of metric spaces

Finite left-distributive algebras and embedding algebras\endtitle

We consider algebras with one binary operation $\cdot$ and one generator ({\it monogenic}) and satisfying the left distributive law $a\cdot (b\cdot c)=(a\cdot b)\cdot (a\cdot c)$. One can define a sequence of finite left-distributive algebras $A_n$, and then take a limit to get an infinite monogenic left-distributive algebra~$A_\infty$. Results of Laver and Steel assuming a strong large cardinal axiom imply that $A_\infty$ is free; it is open whether the freeness of $A_\infty$ can be proved without the large cardinal assumption, or even in Peano arithmetic. The main result of this paper is the equivalence of this problem with the existence of a certain algebra of increasing functions on natural numbers, called an {\it embedding algebra}. Using this and results of the first author, we conclude that the freeness of $A_\infty$ is unprovable in primitive recursive arithmetic

On disjoint Borel uniformizations

Larman showed that any closed subset of the plane with uncountable vertical cross-sections has aleph_1 disjoint Borel uniformizing sets. Here we show that Larman's result is best possible: there exist closed sets with uncountable cross-sections which do not have more than aleph_1 disjoint Borel uniformizations, even if the continuum is much larger than aleph_1. This negatively answers some questions of Mauldin. The proof is based on a result of Stern, stating that certain Borel sets cannot be written as a small union of low-level Borel sets. The proof of the latter result uses Steel's method of forcing with tagged trees; a full presentation of this method, written in terms of Baire category rather than forcing, is given here