Searcharxiv⌕ Search

arXiv subjects

Ben Lund

Publications and source records attributed to Ben Lund.

At least 37 records · Page 2Linked to original sources
arXiv2021

Simple proofs for Furstenberg sets over finite fields

A $(k,m)$-Furstenberg set $S \subset \mathbb{F}_q^n$ over a finite field is a set that has at least $m$ points in common with a $k$-flat in every direction. The question of determining the smallest size of such sets is a natural generalization of the finite field Kakeya problem. The only previously known bound for these sets is due to Ellenberg-Erman and requires sophisticated machinery from algebraic geometry. In this work we give new, completely elementary and simple, proofs which significantly improve the known bounds. Our main result relies on an equivalent formulation of the problem using the notion of min-entropy, which could be of independent interest.

math.CO↗
arXiv2020

On the list recoverability of randomly punctured codes

We show that a random puncturing of a code with good distance is list recoverable beyond the Johnson bound. In particular, this implies that there are Reed-Solomon codes that are list recoverable beyond the Johnson bound. It was previously known that there are Reed-Solomon codes that do not have this property. As an immediate corollary to our main theorem, we obtain better degree bounds on unbalanced expanders that come from Reed-Solomon codes.

math.CO↗
arXiv2020

Two theorems on point-flat incidences

We improve the theorem of Beck giving a lower bound on the number of $k$-flats spanned by a set of points in real space, and improve the bound of Elekes and Tóth on the number of incidences between points and $k$-flats in real space.

math.CO↗
arXiv2019

Perfect matchings and derangements on graphs

We show that each perfect matching in a bipartite graph $G$ intersects at least half of the perfect matchings in $G$. This result has equivalent formulations in terms of the permanent of the adjacency matrix of a graph, and in terms of derangements and permutations on graphs. We give several related results and open questions.

math.CO↗
arXiv2019

A refined energy bound for perpendicular bisectors

Let $\mathcal{P}$ be a set of $n$ points in the Euclidean plane. We prove that, for any $ε> 0$, either a single line or circle contains $n/2$ points of $\mathcal{P}$, or the number of distinct perpendicular bisectors determined by pairs of points in $\mathcal{P}$ is $Ω(n^{52/35 - ε})$, where the constant implied by the $Ω$ notation depends on $\mathcal{P}$. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains $n/2$ points of $\mathcal{P}$, or the number of distinct perpendicular bisectors is $Ω(n^2)$. The proof relies bounding the size of a carefully selected subset of the quadruples $(a,b,c,d) \in \mathcal{P}^4$ such that the perpendicular bisector of $a$ and $b$ is the same as the perpendicular bisector of $c$ and $d$.

math.CO↗
arXiv2019

Finite field Kakeya and Nikodym sets in three dimensions

We give improved lower bounds on the size of Kakeya and Nikodym sets over $\mathbb{F}_q^3$. We also propose a natural conjecture on the minimum number of points in the union of a not-too-flat set of lines in $\mathbb{F}_q^3$, and show that this conjecture implies an optimal bound on the size of a Nikodym set.

math.CO↗
arXiv2018

A Construction for Difference Sets with Local Properties

We construct finite sets of real numbers that have a small difference set and strong local properties. In particular, we construct a set $A$ of $n$ real numbers such that $|A-A|=n^{\log_2 3}$ and that every subset $A'\subseteq A$ of size $k$ satisfies $|A'-A'|\ge k^{\log_2 3}$. This construction leads to the first non-trivial upper bound for the problem of distinct distances with local properties.

math.CO↗
arXiv2018

Bisectors and pinned distances

We prove, under suitable conditions, a lower bound on the number of pinned distances determined by small subsets of two-dimensional vector spaces over fields. For finite subsets of the Euclidean plane we prove an upper bound for their bisector energy.

math.CO↗
arXiv2018

Convex cones, integral zonotopes, limit shape

This paper is about integral zonotopes. It is proven that large zonotopes in a convex cone have a limit shape, meaning that, after suitable scaling, the overwhelming majority of the zonotopes are very close to a fixed convex set. Several combinatorial properties of large zonotopes are established.

math.CO↗
arXiv2018

Distinct spreads in vector spaces over finite fields

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.

math.CO↗
arXiv2016

Incidence Bounds for Block Designs

We prove three theorems giving extremal bounds on the incidence structures determined by subsets of the points and blocks of a balanced incomplete block design (BIBD). These results generalize and strengthen known bounds on the number of incidences between points and $m$-flats in affine geometries over finite fields. First, we show an upper bound on the number of incidences between sufficiently large subsets of the points and blocks of a BIBD. Second, we show that a sufficiently large subset of the points of a BIBD determines many t-rich blocks. Third, we show that a sufficiently large subset of the blocks of a BIBD determines many t-rich points. These last two results are new even in the special case of incidences between points and $m$-flats in an affine geometry over a finite field. As a corollary we obtain a tight bound on the number of t-rich points determined by a set of points in a plane over a finite field, and use it to sharpen a result of Iosevich, Rudnev, and Zhai on the number of triangles with distinct areas determined by a set of points in a plane over a finite field.

math.CO↗
arXiv2016

The sign-sequence constant of the plane

Let $L$ be a finite-dimensional real normed space, and let $B$ be the unit ball in $L$. The sign sequence constant of $L$ is the least $t>0$ such that, for each sequence $v_1, \ldots, v_n \in B$, there are signs $\varepsilon_1, \ldots, \varepsilon_n \in \{-1, +1\}$ such that $\varepsilon_1 v_1 + \ldots + \varepsilon_k v_k \in t B$, for each $1 \leq k \leq n$. We show that the sign sequence constant of a plane is at most $2$, and the sign sequence constant of the plane with the Euclidean norm is equal to $\sqrt{3}$.

math.MG↗
arXiv2016

Essential dimension and the flats spanned by a point set

Let $P$ be a finite set of points in $\mathbb{R}^d$ or $\mathbb{C}^d$. We answer a question of Purdy on the conditions under which the number of hyperplanes spanned by $P$ is at least the number of $(d-2)$-flats spanned by $P$. In answering this question, we define a new measure of the degeneracy of a point set with respect to affine subspaces, termed the "essential dimension". We use the essential dimension to give an asymptotic expression for the number of $k$-flats spanned by $P$, for $1 \leq k \leq d-1$.

math.CO↗
arXiv2016

An improved bound on $(A+A)/(A+A)$

We show that, for a finite set $A$ of real numbers, the size of the set $$\frac{A+A}{A+A} = \left\{ \frac{a+b}{c+d} : a,b,c,d \in A, c+d \neq 0 \right \}$$ is bounded from below by $$\left|\frac{A+A}{A+A} \right| \gg \frac{|A|^{2+1/4}}{|A / A|^{1/8} \log |A|}.$$ This improves a result of Roche-Newton.

math.CO↗
arXiv2016

On distinct perpendicular bisectors and pinned distances in finite fields

Given a set of points $P \subset \mathbb F_q^2$ such that $|P|\geq q^{3/2}$ it is established that $|P|$ determines $Ω(q^2)$ distinct perpendicular bisectors. It is also proven that, if $|P| \geq q^{4/3}$, then for a positive proportion of points $a \in P$, we have $$|\{\| a- b\|: b \in P\}|=Ω(q),$$ where $\|a- b\|$ is the distance between points $a$ and $b$. The latter result represents an improvement on a result of Chapman et al. (arxiv:0903.4218).

math.CO↗
arXiv2015

Incidences and pairs of dot products

Let $\mathbb{F}$ be a field, let $P \subseteq \mathbb{F}^d$ be a finite set of points, and let $α,β\in \mathbb{F} \setminus \{0\}$. We study the quantity \[|Π_{α, β}| = \{(p,q,r) \in P \times P \times P \mid p \cdot q = α, p \cdot r = β\}.\] We observe a connection between the question of placing an upper bound on $|Π_{α,β}|$ and a well-studied question on the number of incidences betwen points and hyperplanes, and use this connection to prove new and strengthened upper bounds on $|Π_{α,β}|$ in a variety of settings.

math.CO↗
arXiv2014

Bisector energy and few distinct distances

We introduce the bisector energy of an $n$-point set $P$ in $\mathbb{R}^2$, defined as the number of quadruples $(a,b,c,d)$ from $P$ such that $a$ and $b$ determine the same perpendicular bisector as $c$ and $d$. If no line or circle contains $M(n)$ points of $P$, then we prove that the bisector energy is $O(M(n)^{\frac{2}{5}}n^{\frac{12}{5}+ε} + M(n)n^2).$. We also prove the lower bound $Ω(M(n)n^2)$, which matches our upper bound when $M(n)$ is large. We use our upper bound on the bisector energy to obtain two rather different results: (i) If $P$ determines $O(n/\sqrt{\log n})$ distinct distances, then for any $0<α\le 1/4$, either there exists a line or circle that contains $n^α$ points of $P$, or there exist $Ω(n^{8/5-12α/5-ε})$ distinct lines that contain $Ω(\sqrt{\log n})$ points of $P$. This result provides new information on a conjecture of Erdős regarding the structure of point sets with few distinct distances. (ii) If no line or circle contains $M(n)$ points of $P$, then the number of distinct perpendicular bisectors determined by $P$ is $Ω(\min\{M(n)^{-2/5}n^{8/5-ε}, M(n)^{-1} n^2\})$. This appears to be the first higher-dimensional example in a framework for studying the expansion properties of polynomials and rational functions over $\mathbb{R}$, initiated by Elekes and Rónyai.

math.CO↗