Abstracts of my papers

1. "O dualności Erdosa-Sierpińskiego dla miary Lebesgue'a i kategorii Baire'a"

   Master's Thesis written under supervision of prof. Piotr Zakrzewski, Warsaw University, 2000 (in Polish).
   
   Abstract. The thesis contains a proof of the fact that the Erdos-Sierpiński mapping from R to R cannot be additive. We also introduce the notion of a very meager set which arises naturally as an analogue of the notion of strong measure zero set but, for some reasons, may be considered a better analogue than the notion of strongly meager set.
   
   Remark: Most of the material contained there can also be found in my other papers.
   
   prac_mag.pdf
   
   

2. "Własności ideałów miary Lebesgue'a i kategorii Baire'a"

   Ph. D. Thesis written under supervision of prof. Piotr Zakrzewski, Polish Academy of Sciences, 2004  (in Polish).
   
   The thesis consists of three main chapters, which are based on the papers "Nonmeaurable algebraic sums of sets of reals", "Small subsets of the reals and tree forcing notions" and "Productivity versus Weak Fubini Property".
   
   doktorat.pdf
   
   

3. "Another note on duality between measure and category"

   Bulletin of the Polish Academy of Sciences, 51 (2003), no. 3,  269-281.
   
   Abstract. We show that no Erdos-Sierpinski mapping from R to R can be additive.
   
   another.pdf
   
   

4. "Small subsets of the reals and tree forcing notions" (with Tomasz Weiss).

   Proceedings of the American Mathematical Society, 132 (2004), 251-259.
   
   Abstract. We discuss the question which properties of smallness in the sense of measure and category (e.g. being universally null, perfectly meager or strongly null set) imply the properties of smallness related to some tree forcing notions (e.g. the properties of being Miller-null or Laver-null).
   
   treeforc.pdf
   
   

5. "Nonmeasurable algebraic sums of sets of reals"

   Colloquium Mathematicum, 102 (2005), no. 1, 113 - 122.
   
   Abstract. We present a theorem which generalizes some known theorems on the existence of nonmeasurable (in various senses) sets of the form X+Y. Some additional related questions concerning measure, category and the algebra of Borel sets are also studied.
   
   nonmeas.pdf
   
   

6. "Set-theoretic properties of Schmidt's ideal" (with Enrico Zoli).

   Georgian Mathematical Journal, 12 (2005), no. 3, 493-504.
   
   Abstract. We study some set-theoretic properties of Schmidt's sigma-ideal on R, emphasizing its analogies and dissimilarities with both the classical sigma-ideals on R of  Lebesgue measure zero sets  and of  Baire first category sets. We highlight the strict analogy between Schmidt's ideal on R and Mycielski's ideal on 2^N.
   
   schmidt.pdf
   

7. "A note on transitive sets without the foundation axiom"

   Reports on Mathematical Logic 40 (2006), 159-163.
   
   Abstract. We construct a model of set theory without the foundation axiom in which there exists a transitive set whose intersection is not transitive.
   
   inttrans.pdf
   
   

8. "Some remarks on indicatrices of measurable functions"

   Bull. Polish Acad. Sci. Math. 53 (2005), 281-284.
   
   Abstract. We show that for a wide class of sigma-algebras A, indicatrices of A-measurable functions admit the same
   characterization as indicatrices of Lebesgue-measurable functions. In particular, this applies to functions measurable in the sense
   of Marczewski.
   
   indicatr.pdf
   

9. "Productivity versus Weak Fubini Property" (with Jan Kraszewski)

   Acta Mathematica Hungarica 111 (2006), 347-353.
   
   Abstract. We construct a sigma-ideal of subsets of the Cantor space which is productive but does not have the Weak Fubini Property. In the construction we use a combinatorial lemma which is of its own interest.
   
   prod_wfp.pdf
   
   
   

10. "Special subsets of the reals and tree forcing notions" (with Andrzej Nowik and Tomasz Weiss)
    

    Proceedings of the American Mathematical Society,  135(2007), no. 9, 2975-2982
    
    Abstract. We study relationships between classes of special subsets of the reals (e.g. meager-additive sets, \gamma-sets, C"-sets, \lambda-sets) and the ideals related to the forcing notions of Laver, Mathias, Miller and Silver.
    
    special.pdf
    
    

11. "Bernstein sets with algebraic properties"
    

    Central European Journal of Mathematics.
    
    Abstract. We construct Bernstein sets in R having some additional algebraic properties. In particular, solving a problem of Kraszewski, Rałowski, Szczepaniak and Żeberski, we construct a Bernstein set which is a < c-covering and improve some other results of Rałowski, Szczepaniak and Żeberski on nonmeasurable sets.
    
    bernstein.pdf