arXiv subjects

Bruno Predojević

Publications and source records attributed to Bruno Predojević.

3 recordsLinked to original sources

arXiv2025

Joint upper Banach density, VC dimensions and Euclidean point configurations

We study two related quantities which generalize the concept of upper Banach density of a set to two measurable subsets of the plane. The first of them allows us to generalize a classic result on sufficiently large distances realized in a set of positive upper density, to distances between points of two sets satisfying an appropriate density condition. The second one allows us to show that for all sufficiently large scales $t>0$ and for a smooth, closed, centrally symmetric, planar curve $\Gamma$ which bounds a convex and compact region in the plane and is of non-vanishing curvature, the family consisting of portions of translates of $t\Gamma$ has the maximal possible Vapnik--Chervonenkis dimension.

math.CA↗

arXiv2024

Polygons of unit area with vertices in sets of infinite planar measure

Paul Erd\H{o}s and R. Daniel Mauldin asked a series of questions on certain types of polygons of area $1$, the vertices of which can be found in every planar set of infinite Lebesgue measure. We address two of these questions, one on cyclic quadrilaterals and the other on convex polygons with congruent sides, with respectively positive and negative answers.

math.CA↗

arXiv2024

Large dilates of hypercube graphs in the plane

We study a distance graph $Γ_n$ that is isomorphic to the $1$-skeleton of an $n$-dimensional unit hypercube. We show that every measurable set of positive upper Banach density in the plane contains all sufficiently large dilates of $Γ_n$. This provides the first examples of distance graphs other than the trees for which a dimensionally sharp embedding in positive density sets is known.

math.CA↗