arXiv subjects

Jonathan Passant

Publications and source records attributed to Jonathan Passant.

6 recordsLinked to original sources

arXiv2025

Generalised Erd\H{o}s distance theory on graphs

The famous Erd\H{o}s distinct distances problem asks the following: how many distinct distances must exist between a set of $n$ points in the plane? There are many generalisations of this question that ask one to consider different spaces and metrics, or larger structures of points. We bring these problems into a common framework using the concept of $g$-rigidity. Specifically, if $G=(V,E)$ is a (hyper)graph, $g$ is a map assigning polynomial measurements to the edges of $G$ and $f_{g,G}(P^V)$ gives the set of $g$-distinct realisations of the $g$-rigid graph $G$, where vertices must lie in a point set $P$, our main results describe sharp lower bounds for the size of $\big|f_{g,G}(P^V)\big|$. This allows us to obtain results for pseudo-Euclidean metrics, $\ell_p$ metrics, dot-product problems, matrix completion problems, and symmetric tensor completion problems. In addition, we use the recent work of Alon, Buci\'c and Sauermann along with a simple colouring argument to prove that the number of $\| \cdot\|$-distinct realisations of a graph $G=(V,E)$ within a $d$-dimensional point set $P$ is at least $\Omega\left(\frac{|P|^{|V|-1}}{(\log |P|)^2} \right)$ for almost all $d$-norms. Our methods here also provide a short proof that the unit distance conjecture implies the pinned distance conjecture.

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arXiv2024

On Distinct Angles in the Plane

We prove that if $N$ points lie in convex position in the plane then they determine $\Omega(N^{5/4})$ distinct angles, provided that the points do not lie on a common circle. This is derived from a more general claim that if $N$ points in the convex position in the real plane determine $KN$ distinct angles, then $K=\Omega(N^{1/4})$ or $\Omega(N/K)$ points are co-circular. The proof makes use of the implicit order one can give to points in convex position and relies on a slightly more general order assumption. The assumption enables one to reduce the issue to counting incidences between points and a multiset of cubic curves, with special attention being paid to the case when the curves are reducible.

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arXiv2022

A Structural Theorem for Sets With Few Triangles

We show that if a finite point set $P\subseteq \mathbb{R}^2$ has the fewest congruence classes of triangles possible, up to a constant $M$, then at least one of the following holds. (1) There is a $\sigma>0$ and a line $l$ which contains $\Omega(|P|^\sigma)$ points of $P$. Further, a positive proportion of $P$ is covered by lines parallel to $l$ each containing $\Omega(|P|^\sigma)$ points of $P$. (2) There is a circle $\gamma$ which contains a positive proportion of $P$. This provides evidence for two conjectures of Erd\H{o}s. We use the result of Petridis-Roche-Newton-Rudnev-Warren on the structure of the affine group combined with classical results from additive combinatorics.

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arXiv2020

On Erd\H{o}s Chains in the Plane

Let $P$ be a finite point set in $\mathbb{R}^2$ with the set of distance $n$-chains defined as $$ \Delta_n(P)=\{(|p_1-p_2|,|p_2-p_3|,\ldots,|p_n-p_{n+1}|):p_i \in P\}.$$ We show that for $2\leq n=O_{|P|}(1)$ we have $$|\Delta_n(P)|\gtrsim \frac{|P|^{n}}{\log^{\frac{13}{2}(n-1)}|P|}.$$ Our argument uses the energy construction of Elekes and a general version of Rudnev's rich-line bound implicit in Rudnev's recent hinge paper which allows one to iterate efficiently on highly intersecting nested subsets of Guth-Katz lines. Let $G$ is a simple connected graph on $m=O(1)$ vertices with $m\geq 2$. Define the graph-distance set $\Delta_G(P)$ as $$ \Delta_G(P) = \{ (|p_{i}-p_{j}|)_{\{i,j\}\in E(G)} : p_i,p_j \in P\}.$$ Combining with results of Guth and Katz and Rudnev with the above, if $G$ has a Hamiltonian path we have $$ |\Delta_G(P)| \gtrsim \frac{|P|^{m-1}}{\text{polylog}|P|}. $$ \end{abstract}

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arXiv2020

Distinct Distances Between a Circle and a Generic Set

Let $S$ be a set of points in $\mathbb{R}^2$ contained in a circle and $P$ an unrestricted point set in $\mathbb{R}^2$. We prove the number of distinct distances between points in $S$ and points in $P$ is at least $\min(|S||P|^{1/4-\varepsilon},|S|^{2/3}|P|^{2/3},|S|^2,|P|^2)$. This builds on work of Pach and De Zeeuw, Bruner and Sharir, McLaughlin and Omar and Mathialagan on distances between pairs of sets.

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arXiv2017

A multi-parameter variant of the Erd\H{o}s distance problem

We study the following variant of the Erd\H{o}s distance problem. Given $E$ and $F$ a point sets in $\mathbb{R}^d$ and $p = (p_1, \ldots, p_q)$ with $p_1+ \cdots + p_q = d$ is an increasing partition of $d$ define $$ B_p(E,F)=\{(|x_1-y_1|, \ldots, |x_q-y_q|): x \in E, y \in F \},$$ where $x=(x_1, \ldots, x_q)$ with $x_i$ in $\mathbb{R}^{p_i}$. For $p_1 \geq 2$ it is not difficult to construct $E$ and $F$ such that $|B_{p}(E,F)|=1$. On the other hand, it is easy to see that if $\gamma_q$ is the best know exponent for the distance problem in $\mathbb{R}^{p_i}$ that $|B_p(E,E)| \geq C{|E|}^{\frac{\gamma_q}{q}}$. The question we study is whether we can improve the exponent $\frac{\gamma_q}{q}$. We first study partitions of length two in detail and prove the optimal result (up to logarithms) that $$ |B_{2,2}(E)| \gtrapprox |E|.$$ In the generalised two dimensional case for $B_{k,l}$ we need the stronger condition that $E$ is $s$-adaptable for $s<\frac{k}{2}+\frac{1}{3}$, letting $\gamma_m$ be the best known exponent for the Erd\H{o}s-distance problem in $\mathbb{R}^m$ for $k \neq l$ we gain a further optimal result of, $$ |B_{k,l}(E)| \gtrapprox |E|^{\gamma_l}.$$ When $k=l$ we use the explicit $\gamma_m=\frac{m}{2}-\frac{2}{m(m+2)}$ result due to Solymosi and Vu to gain $$ |B_{k,k}(E)| \gtrapprox |E|^{\frac{13}{14}\gamma_k}.$$ For a general partition, let $\gamma_i = \frac{2}{p_i}-\frac{2}{p_i(p_i+2)}$ and $\eta_i = \frac{2}{2d-(p_i-1)}$. Then if $E$ is $s$-adaptable with $s>d-\frac{p_1}{2}+\frac{1}{3}$ we have $$ B_p(E) \gtrapprox |E|^\tau \hspace{0.5cm} \text{where} \hspace{0.5cm} \tau = \gamma_q\left(\frac{\gamma_1+\eta_1}{\gamma_q+(q-1)(\gamma_1+\eta_1)}\right).$$ Where $p_i \sim \frac{d}{q}$ implies $\tau \sim \gamma_{q}\left(\frac{1}{q}+\frac{1}{dq}\right)$ and $p_q \sim d$ (with $q<<d$) implies $\tau \sim \gamma_{q}\left(\frac{1}{q}+\frac{1}{q^2}\right)$.

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