Falilies of Trigonometric Thin Sets and Related Exceptional Sets
by
Lev Bukovský
Institute of Mathematics, P. J. Šafárik University, Košice, Slovakia

A family F Í P(á0, 1ñ) is a family of thin sets if ÈF=á0, 1ñ, B Î F provided B Í A Î F and no non-trivial open interval belongs to F. A set A is permitted for family F if AÈB Î F for any B Î F.

I give a brief survey of families of thin sets related to trigonometric series and their basic properties. Replacing the sinus function in definitions by a continuous function we obtain another families of thin sets. Some relationships to corresponding trigonometric families will be presented.

There was an old problem about existence of permitted sets for trigonometric families of thin sets with a perfect subsets or at least of cardinality continuum. I present recent results showing that in the most important cases the answer is undecidable in ZFC. Main tools for the results are sets with some covering properties.

Date received: November 27, 2005