Rafał Filipów

On ideal equal convergence

We consider ideal equal convergence of a sequence of functions. This is a generalization of equal convergence introduced by Császár and Laczkovich [Császár Á., Laczkovich M., Discrete and equal convergence, Studia Sci. Math. Hungar., 1975, 10(3–4), 463–472]. Our definition of ideal equal convergence encompasses two different kinds of ideal equal convergence introduced in [Das P., Dutta S., Pal S.K., On and *-equal convergence and an Egoroff-type theorem, Mat. Vesnik, 2014, 66(2), 165–177]_and [Filipów R., Szuca P., Three kinds of convergence and the associated I-Baire classes, J. Math. Anal. Appl., 2012, 391(1), 1–9]. We also solve a few problems posed in the paper by Das, Dutta and Pal.

Density versions of Schur's theorem for ideals generated by submeasures

We characterize ideals of subsets of natural numbers for which some versions of Schurs theorem hold. These are similar to generalizations shown by Bergelson (1986) in [1] and Frankl, Graham and Rodl (1990) in [7]. Additionally, we prove a generalization of an iterated version of Ramseys theorem.

On the Difference Property of Borel Measurable and \(s\)-Measurable Functions

We prove that the class of (s)-measurable functions does not have the difference property. We show also under CH that there is a function with Borel differences but of unlimited Baire class. It solves a problem of M. Laczkovich.

Coincidence of PTPN22 c.1858CC and FCRL3 ‐169CC genotypes as a biomarker of preserved residual β‐cell function in children with type 1 diabetes

Genotype‐phenotype studies in type 1 diabetes (T1DM) patients are needed for further development of therapy strategies.

On some properties of Hamel bases and their applications to Marczewski measurable functions

We introduce new properties of Hamel bases. We show that it is consistent with ZFC that such Hamel bases exist. Under the assumption that there exists a Hamel basis with one of these properties we construct a discontinuous and additive function that is Marczewski measurable. Moreover, we show that such a function can additionally have the intermediate value property (and even be an extendable function). Finally, we examine sums and limits of such functions.