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On the 1/2 Problem of Besicovitch: quasi-arcs do not contain sharp saw-teeth.

Hany M. Farag (2002)

Revista Matemática Iberoamericana

In this paper we give an alternative proof of our recent result that totally unrectifiable 1-sets which satisfy a measure-theoretic flatness condition at almost every point and sufficiently small scales, satisfy Besicovitch's 1/2-Conjecture which states that the lower spherical density for totally unrectifiable 1-sets should be bounded above by 1/2 at almost every point. This is in contrast to rectifiable 1-sets which actually possess a density equal to unity at almost every point. Our present method...

On the complexity of some σ -ideals of σ -P-porous sets

Luděk Zajíček, Miroslav Zelený (2003)

Commentationes Mathematicae Universitatis Carolinae

Let 𝐏 be a porosity-like relation on a separable locally compact metric space E . We show that the σ -ideal of compact σ - 𝐏 -porous subsets of E (under some general conditions on 𝐏 and E ) forms a Π 1 1 -complete set in the hyperspace of all compact subsets of E , in particular it is coanalytic and non-Borel. Our general results are applicable to most interesting types of porosity. It is shown in the cases of the σ -ideals of σ -porous sets, σ - 〈 g 〉 -porous sets, σ -strongly porous sets, σ -symmetrically porous sets...

On the complexity of sums of Dirichlet measures

Sylvain Kahane (1993)

Annales de l'institut Fourier

Let M be the set of all Dirichlet measures on the unit circle. We prove that M + M is a non Borel analytic set for the weak* topology and that M + M is not norm-closed. More precisely, we prove that there is no weak* Borel set which separates M + M from D ⊥ (or even L 0 ⊥ ) , the set of all measures singular with respect to every measure in M . This extends results of Kaufman, Kechris and Lyons about D ⊥ and H ⊥ and gives many examples of non Borel analytic sets.

On the difference property of Borel measurable functions

Hiroshi Fujita, Tamás Mátrai (2010)

Fundamenta Mathematicae

If an atomlessly measurable cardinal exists, then the class of Lebesgue measurable functions, the class of Borel functions, and the Baire classes of all orders have the difference property. This gives a consistent positive answer to Laczkovich's Problem 2 [Acta Math. Acad. Sci. Hungar. 35 (1980)]. We also give a complete positive answer to Laczkovich's Problem 3 concerning Borel functions with Baire-α differences.

On the difference property of families of measurable functions

Rafał Filipów (2003)

Colloquium Mathematicae

We show that, generally, families of measurable functions do not have the difference property under some assumption. We also show that there are natural classes of functions which do not have the difference property in ZFC. This extends the result of Erdős concerning the family of Lebesgue measurable functions.

On the ideal (v 0)

Piotr Kalemba, Szymon Plewik, Anna Wojciechowska (2008)

Open Mathematics

The σ-ideal (v 0) is associated with the Silver forcing, see [5]. Also, it constitutes the family of all completely doughnut null sets, see [9]. We introduce segment topologies to state some resemblances of (v 0) to the family of Ramsey null sets. To describe add(v 0) we adopt a proof of Base Matrix Lemma. Consistent results are stated, too. Halbeisen’s conjecture cov(v 0) = add(v 0) is confirmed under the hypothesis t = min{cf(c), r}. The hypothesis cov(v 0) = ω 1 implies that (v 0) has the ideal...

On the structure of the intersection of two middle third Cantor sets.

Gregory J. Davis, Tian You Hu (1995)

Publicacions Matemàtiques

Motivated by the study of planar homoclinic bifurcations, in this paper we describe how the intersection of two middle third Cantor sets changes as the sets are translated across each other. The resulting description shows that the intersection is never empty; in fact, the intersection can be either finite or infinite in size. We show that when the intersection is finite then the number of points in the intersection will be either 2n or 3 · 2n. We also explore the Hausdorff dimension of the intersection...

On topologies generated by some operators

Katarzyna Flak, Jacek Hejduk (2013)

Open Mathematics

The paper concerns topologies introduced in a topological space (X, τ) by operators which are much weaker than the lower density operators. Some properties of the family of sets having the Baire property and the family of meager sets with respect to such topologies are investigated.

Currently displaying 61 – 80 of 88