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Ethan Patrick White

Publications and source records attributed to Ethan Patrick White.

10 recordsLinked to original sources
arXiv2025

Improving Uniquely Decodable Codes in Binary Adder Channels

We present a general method to modify existing uniquely decodable codes in the $T$-user binary adder channel. If at least one of the original constituent codes does not have average weight exactly half of the dimension, then our method produces a new set of constituent codes in a higher dimension, with a strictly higher rate. Using our method we improve the highest known rate for the $T$-user binary adder channel for all $T \geq 2$. This information theory problem is equivalent to co-Sidon problems initiated by Lindstr{ö}m in the 1960s, and also the multi-set union-free problem. Our results improve the known lower bounds in these settings as well.

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arXiv2024

An annotated bibliography for comparative prime number theory

The goal of this annotated bibliography is to record every publication on the topic of comparative prime number theory (through mid-2024) together with a summary of its results. We use a unified system of notation for the quantities being studied and for the hypotheses under which results are obtained.

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arXiv2024

A note on the no-$(d+2)$-on-a-sphere problem

For fixed $d\geq 3$, we construct subsets of the $d$-dimensional lattice cube $[n]^d$ of size $n^{\frac{3}{d + 1} - o(1)}$ with no $d+2$ points on a sphere or a hyperplane. This improves the previously best known bound of $Ω(n^{\frac{1}{d-1}})$ due to Thiele from 1995.

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arXiv2024

Grid-drawings of graphs in three-dimensions

Using probabilistic methods, we obtain grid-drawings of graphs without crossings with low volume and small aspect ratio. We show that every $D$-degenerate graph on $n$ vertices can be drawn in $[m]^3$ where $m^3 = O(D^2 n\log n)$. In particular, every graph of bounded maximum degree can be drawn in a grid with volume $O(n \log n)$.

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arXiv2023

On Axial Symmetry in Convex Bodies

For a two-dimensional convex body, the Kovner-Besicovitch measure of symmetry is defined as the volume ratio of the largest centrally symmetric body contained inside the body to the original body. A classical result states that the Kovner-Besicovitch measure is at least $2/3$ for every convex body and equals $2/3$ for triangles. Lassak showed that an alternative measure of symmetry, i.e., symmetry about a line (axiality) has a value of at least $2/3$ for every convex body. However, the smallest known value of the axiality of a convex body is around $0.81584$, achieved by a convex quadrilateral. We show that every plane convex body has axiality at least $\frac{2}{41}(10 + 3 \sqrt{2}) \approx 0.69476$, thereby establishing a separation with the central symmetry measure. Moreover, we find a family of convex quadrilaterals with axiality approaching $\frac{1}{3}(\sqrt{2}+1) \approx 0.80474$. We also establish improved bounds for a ``folding" measure of axial symmetry for plane convex bodies. Finally, we establish improved bounds for a generalization of axiality to high-dimensional convex bodies.

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arXiv2023

Additive structure in convex translates

Let $\mathcal{P}$ be a set of points in the plane, and $\mathcal{S}$ a strictly convex set of points. In this note, we show that if $\mathcal{P}$ contains many translates of $\mathcal{S}$, then these translates must come from a generalized arithmetic progression of low dimension. We also discuss an application to the unit distance conjecture.

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arXiv2022

An almost-tight $L^2$ autoconvolution inequality

Let $\mathcal{F}$ denote the set of functions $f \colon [-1/2,1/2] \to \mathbb{R}$ such that $\int f = 1$. We determine the value of $\inf_{f \in \mathcal{F}} \| f \ast f \|_2$ up to a 0.0014\% error, thereby making progress on a problem asked by Ben Green. Furthermore, we prove that a unique minimizer exists. As a corollary, we obtain improvements on the maximum size of $B_h[g]$ sets for $(g,h) \in \{ (2,2),(3,2),(4,2),(1,3),(1,4)\}$.

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arXiv2022

Erdős' minimum overlap problem

We obtain a substantially improved lower bound for the minimum overlap problem asked by Erdős. Our approach uses elementary Fourier analysis to translate the problem to a convex optimization program.

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arXiv2021

Asymptotics for the number of directions determined by $[n] \times [n]$ in $\mathbb{F}_p^2$

Let $p$ be a prime and $n$ a positive integer such that $\sqrt{\frac p2} + 1 \leq n \leq \sqrt{p}$. For any arithmetic progression $A$ of length $n$ in $\mathbb{F}_p$, we establish an asymptotic formula for the number of directions determined by $A \times A \subset \mathbb{F}_p^2$. The key idea is to reduce the problem to counting the number of solutions to the bilinear Diophantine equation $ad+bc=p$ in variables $1\le a,b,c,d\le n$; our asymptotic formula for the number of solutions is of independent interest.

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