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Eyvindur A. Palsson

Publications and source records attributed to Eyvindur A. Palsson.

At least 19 recordsLinked to original sources
arXiv2024

On Optimal Point Sets Determining Distinct Triangles

Erdős and Fishburn studied the maximum number of points in the plane that span $k$ distances and classified these configurations, as an inverse problem of the Erdős distinct distances problem. We consider the analogous problem for triangles. Past work has obtained the optimal sets for one and two distinct triangles in the plane. In this paper, we resolve a conjecture that at most six points in the plane can span three distinct triangles, and obtain the hexagon as the unique configuration that achieves this. We also provide evidence that optimal sets cannot be on the square lattice in the general case.

math.CO↗
arXiv2023

The Triangle Operator

We examine the averaging operator $T$ corresponding to the manifold in $\mathbb{R}^{2d}$ of pairs of points $(u,v)$ satisfying $|u| = |v| = |u - v| = 1$, so that $\{0,u,v\}$ is the set of vertices of an equilateral triangle. We establish $L^p \times L^q \rightarrow L^r$ boundedness for $T$ for $(1/p, 1/q, 1/r)$ in the convex hull of the set of points $\lbrace (0, 0, 0) ,\, (1, 0 , 1) ,\, (0, 1, 1) , \, ({1}/{p_d}, {1}/{p_d}, {2}/{p_d}) \rbrace$, where $p_d = \frac{19d-4}{11d - 12}$ and $d\geq 7$.

math.CA↗
arXiv2023

Distinct Angles and Angle Chains in Three Dimensions

In 1946, Erdős posed the distinct distance problem, which seeks to find the minimum number of distinct distances between pairs of points selected from any configuration of $n$ points in the plane. The problem has since been explored along with many variants, including ones that extend it into higher dimensions. Less studied but no less intriguing is Erdős' distinct angle problem, which seeks to find point configurations in the plane that minimize the number of distinct angles. In their recent paper "Distinct Angles in General Position," Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf use a logarithmic spiral to establish an upper bound of $O(n^2)$ on the minimum number of distinct angles in the plane in general position, which prohibits three points on any line or four on any circle. We consider the question of distinct angles in three dimensions and provide bounds on the minimum number of distinct angles in general position in this setting. We focus on pinned variants of the question, and we examine explicit constructions of point configurations in $\mathbb{R}^3$ which use self-similarity to minimize the number of distinct angles. Furthermore, we study a variant of the distinct angles question regarding distinct angle chains and provide bounds on the minimum number of distinct chains in $\mathbb{R}^2$ and $\mathbb{R}^3$.

cs.CG↗
arXiv2022

Optimal Point Sets Determining Few Distinct Angles

We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most $k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds of $k+2 \leq P(k) \leq 6k$, although the upper bound may be improved pending progress toward the Weak Dirac Conjecture. Notably, it is surprising that $P(k)=Θ(k)$ since, in the distance setting, the best known upper bound on the analogous quantity is quadratic and no lower bound is well-understood.

math.CO↗
arXiv2022

Distinct Angles in General Position

The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is Erdős' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear points in the plane. Recent work has introduced bounds on a wide array of variants of this problem, inspired by similar variants in the distance setting. In this short note, we improve the best known upper bound for the minimum number of distinct angles formed by $n$ points in general position from $O(n^{\log_2(7)})$ to $O(n^2)$. Before this work, similar bounds relied on projections onto a generic plane from higher dimensional space. In this paper, we employ the geometric properties of a logarithmic spiral, sidestepping the need for a projection. We also apply this configuration to reduce the upper bound on the largest integer such that any set of $n$ points in general position has a subset of that size with all distinct angles. This bound is decreased from $O(n^{\log_2(7)/3})$ to $O(n^{1/2})$.

cs.CG↗
arXiv2021

Distinct Angle Problems and Variants

The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Less well known is Erdős' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear points in the plane. The standard problem is already well understood. However, it admits many of the same variants as the distinct distance problem, many of which are unstudied. We provide upper and lower bounds on a broad class of distinct angle problems. We show that the number of distinct angles formed by $n$ points in general position is $O(n^{\log_2(7)})$, providing the first non-trivial bound for this quantity. We introduce a new class of asymptotically optimal point configurations with no four cocircular points. Then, we analyze the sensitivity of asymptotically optimal point sets to perturbation, yielding a much broader class of asymptotically optimal configurations. In higher dimensions we show that a variant of Lenz's construction admits fewer distinct angles than the optimal configurations in two dimensions. We also show that the minimum size of a maximal subset of $n$ points in general position admitting only unique angles is $Ω(n^{1/5})$ and $O(n^{\log_2(7)/3})$. We also provide bounds on the partite variants of the standard distinct angle problem.

math.CO↗
arXiv2019

Crescent configurations in normed spaces

We study the problem of crescent configurations, posed by Erdős in 1989. A crescent configuration is a set of $n$ points in the plane such that: 1) no three points lie on a common line, 2) no four points lie on a common circle, 3) for each $1 \leq i \leq n - 1$, there exists a distance which occurs exactly $i$ times. Constructions of sizes $n \leq 8$ have been provided by Liu, Palásti, and Pomerance. Erdős conjectured that there exists some $N$ for which there do not exist crescent configurations of size $n$ for all $n \geq N$. We extend the problem of crescent configurations to general normed spaces $(\mathbb{R}^2, \| \cdot \|)$ by studying strong crescent configurations in $\| \cdot \|$. In an arbitrary norm $\|\cdot \|$, we construct a strong crescent configuration of size 4. We also construct larger strong crescent configurations in the Euclidean, taxicab, and Chebyshev norms, of sizes $n \leq 6$, $n \leq 8$, and $n \leq 8$ respectively. When defining strong crescent configurations, we introduce the notion of line-like configurations in $\|\cdot \|$. A line-like configuration in $\|\cdot \|$ is a set of points whose distance graph is isomorphic to the distance graph of equally spaced points on a line. In a broad class of norms, we construct line-like configurations of arbitrary size. Our main result is a crescent-type result about line-like configurations in the Chebyshev norm. A line-like crescent configuration is a line-like configuration for which no three points lie on a common line and no four points lie on a common $\|\cdot \|$ circle. We prove that for $n \geq 7$, every line-like crescent configuration of size $n$ in the Chebyshev norm must have a rigid structure. Specifically, it must be a perpendicular perturbation of equally spaced points on a horizontal or vertical line.

math.CO↗
arXiv2019

An improved dimensional threshold for the angle problem

The Falconer distinct distance problem asks for a compact set $E\subset\mathbb{R}^d$ how large its Hausdorff dimension needs to be to ensure that the Lebesgue measure of its distance set is positive. In this paper we consider the analogous question for the set of angles. We show that if the Hausdorff dimension of $E$ is strictly bigger than $\frac{d}{2}$ then the Lebesgue measure of the angles set is positive. In the plane this result was previously established by Harangi et al. In higher dimensions, our exponent improves the $\frac{d+1}{2}$ threshold previously obtain by the authors of this paper and Mihalis Mourgoglou. We do not know what the right dimensional threshold should be in higher dimensions.

math.CA↗
arXiv2019

Characterizing optimal point sets determining one distinct triangle

In this paper we determine the maximum number of points in $\mathbb{R}^d$ which form exactly $t$ distinct triangles, where we restrict ourselves to the case of $t = 1$. We denote this quantity by $F_d(t)$. It was known from the work of Epstein et al. that $F_2(1) = 4$. Here we show somewhat surprisingly that $F_3(1) = 4$ and $F_d(1) = d + 1$, whenever $d \geq 3$, and characterize the optimal point configurations. This is an extension of a variant of the distinct distance problem put forward by Erdős and Fishburn.

math.CO↗
arXiv2019

Uniqueness of Optimal Point Sets Determining Two Distinct Triangles

In this paper, we show that the maximum number of points in $d\geq3$ dimensions determining exactly 2 distinct triangles is $2d$. We further show that this maximum is uniquely achieved by the vertices of the $d$-orthoplex. We build upon the work of Hirasaka and Shinohara who determined that the $d$-orthoplex is such an optimal configuration, but did not prove its uniqueness. Further, we present a more elementary argument for its optimality.

math.CO↗
arXiv2019

Classification of crescent configurations

Let $n$ points be in crescent configurations in $\mathbb{R}^d$ if they lie in general position in $\mathbb{R}^d$ and determine $n-1$ distinct distances, such that for every $1 \leq i \leq n-1$ there is a distance that occurs exactly $i$ times. Since Erdős' conjecture in 1989 on the existence of $N$ sufficiently large such that no crescent configurations exist on $N$ or more points, he, Pomerance, and Palásti have given constructions for $n$ up to $8$ but nothing is yet known for $n \geq 9$. Most recently, Burt et. al. had proven that a crescent configuration on $n$ points exists in $\mathbb{R}^{n-2}$ for $n \geq 3$. In this paper, we study the classification of these configurations on $4$ and $5$ points through graph isomorphism and rigidity. Our techniques, which can be generalized to higher dimensions, offer a new viewpoint on the problem through the lens of distance geometry and provide a systematic way to construct crescent configurations.

math.CO↗
arXiv2018

On algorithms to calculate integer complexity

We consider a problem first proposed by Mahler and Popken in 1953 and later developed by Coppersmith, Erdős, Guy, Isbell, Selfridge, and others. Let $f(n)$ be the complexity of $n \in \mathbb{Z^{+}}$, where $f(n)$ is defined as the least number of $1$'s needed to represent $n$ in conjunction with an arbitrary number of $+$'s, $*$'s, and parentheses. Several algorithms have been developed to calculate the complexity of all integers up to $n$. Currently, the fastest known algorithm runs in time $\mathcal{O}(n^{1.230175})$ and was given by J. Arias de Reyna and J. van de Lune in 2014. This algorithm makes use of a recursive definition given by Guy and iterates through products, $f(d) + f\left(\frac{n}{d}\right)$, for $d \ |\ n$, and sums, $f(a) + f(n - a)$, for $a$ up to some function of $n$. The rate-limiting factor is iterating through the sums. We discuss potential improvements to this algorithm via a method that provides a strong uniform bound on the number of summands that must be calculated for almost all $n$. We also develop code to run J. Arias de Reyna and J. van de Lune's analysis in higher bases and thus reduce their runtime of $\mathcal{O}(n^{1.230175})$ to $\mathcal{O}(n^{1.222911236})$. All of our code can be found online at: https://github.com/kcordwel/Integer-Complexity.

math.NT↗
arXiv2018

The Inverse Gamma Distribution and Benford's Law

According to Benford's Law, many data sets have a bias towards lower leading digits (about $30\%$ are $1$'s). The applications of Benford's Law vary: from detecting tax, voter and image fraud to determining the possibility of match-fixing in competitive sports. There are many common distributions that exhibit such bias, i.e. they are almost Benford. These include the exponential and the Weibull distributions. Motivated by these examples and the fact that the underlying distribution of factors in protein structure follows an inverse gamma distribution, we determine the closeness of this distribution to a Benford distribution as its parameters change.

math.PR↗
arXiv2017

Dimensional lower bounds for Falconer type incidence and point configuration theorems

Let $1 \leq k \leq d$ and consider a subset $E\subset \mathbb{R}^d$. In this paper, we study the problem of how large the Hausdorff dimension of $E$ must be in order for the set of distinct noncongruent $k$-simplices in $E$ (that is, noncongruent point configurations of $k+1$ points from $E$) to have positive Lebesgue measure. This generalizes the $k=1$ case, the well-known Falconer distance problem and a major open problem in geometric measure theory. We establish a dimensional lower threshold of $\frac{d(k+1)}{d+2}$ for Falconer type theorems for $k$-simplices. This threshold is nontrivial in the range $d/2 \leq k \leq d$ and is obtained through counting simplices in a standard lattice using results of the Gauss circle problem. Many results on Falconer type theorems have been established through incidence theorems, which generally establish sufficient but not necessary conditions for the point configuration theorems. We also establish a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for $k$-simplices where $k\leq d \leq 2k+1$ by generalizing an example of Mattila. Finally, we prove a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for triangles in a convex setting in every dimension greater than $3$. This last result generalizes work by Iosevich and Senger on distances that was built on a construction by Valtr. The final result utilizes number-theoretic machinery to estimate the number of solutions to a Diophantine equation.

math.CA↗
arXiv2017

Optimal point sets determining few distinct triangles

We generalize work of Erdos and Fishburn to study the structure of finite point sets that determine few distinct triangles. Specifically, we ask for a given $t$, what is the maximum number of points that can be placed in the plane to determine exactly $t$ distinct triangles? Denoting this quantity by $F(t)$, we show that $F(1) = 4$, $F(2) = 5$, and $F(t) < 48(t+1)$ for all $t$. We also completely characterize the optimal configurations for $t = 1, 2$.

math.CO↗
arXiv2015

A Generalization of Zeckendorf's Theorem via Circumscribed $m$-gons

Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbers satisfy $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$, $F_1=1$ and $F_2=2$. The distribution of the number of summands in such decomposition converges to a Gaussian, the gaps between summands converges to geometric decay, and the distribution of the longest gap is similar to that of the longest run of heads in a biased coin; these results also hold more generally, though for technical reasons previous work needed to assume the coefficients in the recurrence relation are non-negative and the first term is positive. We extend these results by creating an infinite family of integer sequences called the $m$-gonal sequences arising from a geometric construction using circumscribed $m$-gons. They satisfy a recurrence where the first $m+1$ leading terms vanish, and thus cannot be handled by existing techniques. We provide a notion of a legal decomposition, and prove that the decompositions exist and are unique. We then examine the distribution of the number of summands used in the decompositions and prove that it displays Gaussian behavior. There is geometric decay in the distribution of gaps, both for gaps taken from all integers in an interval and almost surely in distribution for the individual gap measures associated to each integer in the interval. We end by proving that the distribution of the longest gap between summands is strongly concentrated about its mean, behaving similarly as in the longest run of heads in tosses of a coin.

math.NT↗
arXiv2015

Crescent configurations

In 1989, Erdős conjectured that for a sufficiently large $n$ it is impossible to place $n$ points in general position in a plane such that for every $1\le i \le n-1$ there is a distance that occurs exactly $i$ times. For small $n$ this is possible and in his paper he provided constructions for $n\leq 8$. The one for $n=5$ was due to Pomerance while Palásti came up with the constructions for $n=7,8$. Constructions for $n=9$ and above remain undiscovered, and little headway has been made toward a proof that for sufficiently large $n$ no configuration exists. In this paper we consider a natural generalization to higher dimensions and provide a construction which shows that for any given $n$ there exists a sufficiently large dimension $d$ such that there is a configuration in $d$-dimensional space meeting Erdős' criteria.

math.CO↗