For a semi-Pfaffian set, i.e., a real semianalytic set defined by equations and inequalities between Pfaffian functions in an open domain G, the frontier and closure in G are represented as semi-Pfaffian sets. The complexity of this representation is estimated in terms of the complexity of the original set. Introduction. Pfaffian functions introduced by Khovanskii [K] are analytic functions sat...

A semioval in a finite projective plane is a non-empty pointset S with the property that for every point in S there exists a unique line tP such that S ∩ tP = {P}. This line is called the tangent to S at P . Semiovals arise in several parts of finite geometries: as absolute points of a polarity (ovals, unitals), as special minimal blocking sets (vertexless triangle), in connection with cryptogr...

We answer a recent question of Bienvenu, Muchnik, Shen, and Vereshchagin. In particular, we prove an effective version of the standard fact from analysis which says that, for any ε > 0 and any Lebesgue-measurable subset of Cantor space, X ⊆ 2, there is an open set Uε ⊆ 2, Uε ⊇ X, such that μ(Uε) ≤ μ(X) + ε, where μ(Z) denotes the Lebesgue measure of Z ⊆ 2. More specifically, our main result sho...