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Register a new userDistinct Distances with $ell_p$ Spaces
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Authors
Moaaz AlQady
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We study ErdH oss distinct distances problem under $ell_p$ metrics with integer $p$. We improve the current best bound for this problem from $Omega(n^{4/5})$ to $Omega(n^{6/7-epsilon})$, for any $epsilon>0$. We also characterize the sets that span an asymptotically minimal number of distinct distances under the $ell_1$ and $ell_infty$ metrics.
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The emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in $mathcal{O}(log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth 2 and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distance-number. Moreover, as $Delta$ increases the existential lower bound on the distance-number of $Delta$-regular graphs tends to $Omega(n^{0.864138})$.
In this short note, we study the distribution of spreads in a point set $mathcal{P} subseteq mathbb{F}_q^d$, which are analogous to angles in Euclidean space. More precisely, we prove that, for any $varepsilon > 0$, if $|mathcal{P}| geq (1+varepsilon) q^{lceil d/2 rceil}$, then $mathcal{P}$ generates a positive proportion of all spreads. We show that these results are tight, in the sense that there exist sets $mathcal{P} subset mathbb{F}_q^d$ of size $|mathcal{P}| = q^{lceil d/2 rceil}$ that determine at most one spread.
We consider the number of distinct distances between two finite sets of points in ${bf R}^k$, for any constant dimension $kge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O(1)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then $$ Omegaleft(minleft{ n^{2/3}m^{2/3},; frac{n^{10/11}m^{4/11}}{log^{2/11}m},; n^2,; m^2right}right) . $$ Without the assumption on $P_2$, there exist sets $P_1$, $P_2$ as above, with only $O(m+n)$ distinct distances between them.
For any cofinite Fuchsian group $Gammasubset {rm PSL}(2, mathbb{R})$, we show that any set of $N$ points on the hyperbolic surface $Gammabackslashmathbb{H}^2$ determines $geq C_{Gamma} frac{N}{log N}$ distinct distances for some constant $C_{Gamma}>0$ depending only on $Gamma$. In particular, for $Gamma$ being any finite index subgroup of ${rm PSL}(2, mathbb{Z})$ with $mu=[{rm PSL}(2, mathbb{Z}): Gamma ]<infty$, any set of $N$ points on $Gammabackslashmathbb{H}^2$ determines $geq Cfrac{N}{mulog N}$ distinct distances for some absolute constant $C>0$.
Let $X$ be a sequence space and denote by $Z(X)$ the subset of $X$ formed by sequences having only a finite number of zero coordinates. We study algebraic properties of $Z(X)$ and show (among other results) that (for $p in [1,infty]$) $Z(ell_p)$ does not contain infinite dimensional closed subspaces. This solves an open question originally posed by R. M. Aron and V. I. Gurariy in 2003 on the linear structure of $Z(ell_infty)$. In addition to this, we also give a thorough analysis of the existing algebraic structures within the set $X setminus Z(X)$ and its algebraic genericity.
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