Exceptional directions for Sierpiński’s

nonmeasurable sets by B. K i r chh e im (Bratislava) and T. Na tkan i e c (Bydgoszcz) Abstract. In [2] the question was considered in how many directions can a nonmea-surable plane set behave even “better ” than the classical one constructed by Sierpiński in [6], in the sense that any line in a given direction intersects the set in at most one point. We considerably improve these results and give a much sharper estimate for the size of the sets of those “better ” directions. I. Let us establish some terminology to be used later. We shall denote by L the outer Lebesgue measure in the plane R2 and by | · | the outer Lebesgue measure on the real line R. By `(x, y) and `δ(x) we denote the line through the two different points x, y ∈ R2 and the line through x in the direction δ ∈ [0, pi), respectively. For a given line ` ⊂ R2 let dir(`) ∈ [0, pi) be its direction. Further, for each M ⊂ [0, pi) let CM denote the cone generated by M, i.e. CM = ⋃{`δ(0) : δ ∈ M}. For k ≥ 1 and δ ∈ [0, pi) let Gk(δ) be the class of all E ⊂ R2 such that for any line ` with dir(`) = δ the set E ∩ ` is open in ` and has at most k connected components. For any M ⊂ R we denote by BSCL(M) = M ∪ {x: (x − ε, x) ∩M 6= ∅ 6 = (x, x + ε) ∩M for any ε> 0}, the “two-sided ” closure of M. Finally, for A ⊂ R2 let intq(A) denote the interior of A in the qualitative topology, i.e. the union of all subsets of A of the form U \ I, where U is open and I is of the first category. We shall use the following set-theoretical assumptions (see [3] for defi-nitions). Let A(m) and U(m) stand for the propositions that the union of less than c (the continuum) measure zero sets has measure zero and that every set of reals of cardinality less than c has measure zero, respectively. Furthermore, let A(c) and U(c) be defined similarly with “first category” replacing measure zero. It is well known that all these conditions follow from Martin’s Axiom and therefore also from the Continuum Hypothesi