arXiv subjects

Rom Pinchasi

Publications and source records attributed to Rom Pinchasi.

24 records · Page 2Linked to original sources

arXiv2013

The number of distinct distances from a vertex of a convex polygon

Erdős conjectured in 1946 that every n-point set P in convex position in the plane contains a point that determines at least floor(n/2) distinct distances to the other points of P. The best known lower bound due to Dumitrescu (2006) is 13n/36 - O(1). In the present note, we slightly improve on this result to (13/36 + eps)n - O(1) for eps ~= 1/23000. Our main ingredient is an improved bound on the maximum number of isosceles triangles determined by P.

cs.CG↗

arXiv2012

Solving $a\pm b=2c$ in the elements of finite sets

We show that if $A$ and $B$ are finite sets of real numbers, then the number of triples $(a,b,c)\in A\times B\times (A\cup B)$ with $a+b=2c$ is at most $(0.15+o(1))(|A|+|B|)^2$ as $|A|+|B|\to\infty$. As a corollary, if $A$ is antisymmetric (that is, $A\cap(-A)=\est$), then there are at most $(0.3+o(1))|A|^2$ triples $(a,b,c)$ with $a,b,c\in A$ and $a-b=2c$. In the general case where $A$ is not necessarily antisymmetric, we show that the number of triples $(a,b,c)$ with $a,b,c\in A$ and $a-b=2c$ is at most $(0.5+o(1))|A|^2$. These estimates are sharp.

math.NT↗

arXiv2012

A note on coloring line arrangements

We show that the lines of every arrangement of $n$ lines in the plane can be colored with $O(\sqrt{n/ \log n})$ colors such that no face of the arrangement is monochromatic. This improves a bound of Bose et al. \cite{BCC12} by a $Θ(\sqrt{\log n})$ factor. Any further improvement on this bound will improve the best known lower bound on the following problem of Erdős: Estimate the maximum number of points in general position within a set of $n$ points containing no four collinear points.

cs.CG↗

arXiv2011

Ice-Creams and Wedge Graphs

What is the minimum angle $α>0$ such that given any set of $α$-directional antennas (that is, antennas each of which can communicate along a wedge of angle $α$), one can always assign a direction to each antenna such that the resulting communication graph is connected? Here two antennas are connected by an edge if and only if each lies in the wedge assigned to the other. This problem was recently presented by Carmi, Katz, Lotker, and Rosén \cite{CKLR10} who also found the minimum such $α$ namely $α=\fracπ{3}$. In this paper we give a simple proof of this result. Moreover, we obtain a much stronger and optimal result (see Theorem \ref{theorem:main}) saying in particular that one can chose the directions of the antennas so that the communication graph has diameter $\le 4$. Our main tool is a surprisingly basic geometric lemma that is of independent interest. We show that for every compact convex set $S$ in the plane and every $0 < α< π$, there exist a point $O$ and two supporting lines to $S$ passing through $O$ and touching $S$ at two \emph{single points} $X$ and $Y$, respectively, such that $|OX|=|OY|$ and the angle between the two lines is $α$.

cs.CG↗

arXiv2008

Additive Bases in Abelian Groups

Let $G$ be a finite, non-trivial abelian group of exponent $m$, and suppose that $B_1, ..., B_k$ are generating subsets of $G$. We prove that if $k>2m \ln \log_2 |G|$, then the multiset union $B_1\cup...\cup B_k$ forms an additive basis of $G$; that is, for every $g\in G$ there exist $A_1\subset B_1, ..., A_k\subset B_k$ such that $g=\sum_{i=1}^k\sum_{a\in A_i} a$. This generalizes a result of Alon, Linial, and Meshulam on the additive bases conjecture. As another step towards proving the conjecture, in the case where $B_1, ..., B_k$ are finite subsets of a vector space we obtain lower-bound estimates for the number of distinct values, attained by the sums of the form $\sum_{i=1}^k \sum_{a\in A_i} a$, where $A_i$ vary over all subsets of $B_i$ for each $i=1, >..., k$. Finally, we establish a surprising relation between the additive bases conjecture and the problem of covering the vertices of a unit cube by translates of a lattice, and present a reformulation of (the strong form of) the conjecture in terms of coverings.

math.NT↗

arXiv2008

On the maximum size of an anti-chain of linearly separable sets and convex pseudo-discs

We show that the maximum cardinality of an anti-chain composed of intersections of a given set of n points in the plane with half-planes is close to quadratic in n. We approach this problem by establishing the equivalence with the problem of the maximum monotone path in an arrangement of n lines. For a related problem on antichains in families of convex pseudo-discs we can establish the precise asymptotic bound: it is quadratic in n. The sets in such a family are characterized as intersections of a given set of n points with convex sets, such that the difference between the convex hulls of any two sets is nonempty and connected.

math.MG↗