On Density of Ratio Sets of Powers of Primes

Denote by R+ and N the set of all positive real numbers and the natural numbers,respectively. Let P = {p1, . . . , pn, . . . } be the set of all primes enumerated inincreasing order. Denote by R(A, B) = {a b; a ? A, b ? B} the ratio set of A, B ? R+ and put R(A) = R(A, A) for A ? R+ (cf. [3],[4],[5]). Note that R(A, B) 6= R(B, A) in general, however R(A, B) is dense in R+ if and only if R(B, A) is dense in R+

On Polygons Excluding Point Sets

By a polygonization of a finite point set $S$ in the plane we understand a simple polygon having $S$ as the set of its vertices. Let $B$ and $R$ be sets of blue and red points, respectively, in the plane such that $B\cup R$ is in general position, and the convex hull of $B$ contains $k$ interior blue points and $l$ interior red points. Hurtado et al. found sufficient conditions for the existence of a blue polygonization that encloses all red points. We consider the dual question of the existence of a blue polygonization that excludes all red points $R$. We show that there is a minimal number $K=K(l)$, which is polynomial in $l$, such that one can always find a blue polygonization excluding all red points, whenever $k\geq K$. Some other related problems are also considered.Comment: 14 pages, 15 figure

A note on large Kakeya sets

A Kakeya set $\mathcal{K}$ in an affine plane of order $q$ is the point set covered by a set $\mathcal{L}$ of $q+1$ pairwise non-parallel lines. Large Kakeya sets were studied by Dover and Mellinger; in [6] they showed that Kakeya sets with size at least $q^2-3q+9$ contain a large knot (a point of $\mathcal{K}$ lying on many lines of $\mathcal{L}$). In this paper, we improve on this result by showing that Kakeya set of size at least $\approx q^2-q\sqrt{q}+\frac{3}{2}q$ contain a large knot. Furthermore, we obtain a sharp result for planes of square order containing a Baer subplane.Comment: To appear in Advances in Geometr