Joshua Zahl

activity

20062026

most citedSpectral gaps, additive energy, and a fractal uncertainty principle

46 citations · 441 across the 46 of their papers we have counts for

collaborators

Showing math.COShow all

18 papers · 1 filter

math.CO2025

On the number of pairwise touching cylinders in $R^{d}$

Jozsef Solymosi, Josh Zahl

John E. Littlewood posted the question {\em ``Is it possible in 3-space for seven infinite circular cylinders of unit radius each to touch all the others? Seven is the number sugge…

math.CO2022★ 5 cited

Improved Elekes-Szabó type estimates using proximity

Jozsef Solymosi, Joshua Zahl

We prove a new Elekes-Szabó type estimate on the size of the intersection of a Cartesian product $A \times B \times C$ with an algebraic surface ${f = 0}$ over the reals. In particul…

math.CO2020

On rich lenses in planar arrangements of circles and related problems

Esther Ezra, Orit E. Raz, Micha Sharir +1

We show that the maximum number of pairwise non-overlapping $k$ -rich lenses (lenses formed by at least $k$ circles) in an arrangement of $n$ circles in the plane is $O\left(\frac{n…

math.CO2020★ 9 cited

Distinct distances in the complex plane

Adam Sheffer, Joshua Zahl

We prove that if $P$ is a set of $n$ points in $C^{2}$ , then either the points in $P$ determine $Ω (n^{1 - ε})$ complex distances, or $P$ is contained in a line with slope $\pm…

math.CO2020★ 2 cited

Sphere tangencies, line incidences, and Lie's line-sphere correspondence

Joshua Zahl

Two spheres with centers $p$ and $q$ and signed radii $r$ and $s$ are said to be in contact if $∣ p - q ∣^{2} = (r - s)^{2}$ . Using Lie's line-sphere correspondence, we show that if $F$ is a…

math.CO2018★ 1 cited

Counting higher order tangencies for plane curves

Joshua Zahl

We prove that $n$ plane algebraic curves determine $O (n^{(k + 2) / (k + 1)})$ points of $k$ -th order tangency. This generalizes an earlier result of Ellenberg, Solymosi, and Zahl on the…

math.CO2016★ 37 cited

Cutting Algebraic Curves into Pseudo-segments and Applications

Micha Sharir, Joshua Zahl

We show that a set of $n$ algebraic plane curves of constant maximum degree can be cut into $O (n^{3/2} polylog n)$ Jordan arcs, so that each pair of arcs intersect at…

math.CO2015★ 3 cited

Curves in $R^{4}$ and two-rich points

Larry Guth, Joshua Zahl

We obtain a new bound on the number of two-rich points spanned by an arrangement of low degree algebraic curves in $R^{4}$ . Specifically, we show that an arrangement of $n$ …

math.CO2015

New bounds on curve tangencies and orthogonalities

Jordan S. Ellenberg, Jozsef Solymosi, Joshua Zahl

We establish new bounds on the number of tangencies and orthogonal intersections determined by an arrangement of curves. First, given a set of $n$ algebraic plane curves, we show t…

math.CO2015★ 1 cited

A note on rich lines in truly high dimensional sets

Joshua Zahl

We modify an argument of Hablicsek and Scherr to show that if a collection of points in $C^{d}$ spans many $r$ --rich lines, then many of these lines must lie in a common $(d…

math.CO2015★ 12 cited

Point-curve incidences in the complex plane

Adam Sheffer, Endre Szabó, Joshua Zahl

We prove an incidence theorem for points and curves in the complex plane. Given a set of $m$ points in $R^{2}$ and a set of $n$ curves with $k$ degrees of freedom, Pach and…

math.CO2014

A semi-algebraic version of Zarankiewicz's problem

Jacob Fox, János Pach, Adam Sheffer +2

A bipartite graph $G$ is semi-algebraic in $R^{d}$ if its vertices are represented by point sets $P, Q \subset R^{d}$ and its edges are defined as pairs of points $(p,…

math.CO2013★ 2 cited

Few distinct distances implies no heavy lines or circles

Adam Sheffer, Joshua Zahl, Frank de Zeeuw

We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set P of n points determines o(n) distinct distance…

math.CO2012★ 32 cited

Improved bounds for incidences between points and circles

Micha Sharir, Adam Sheffer, Joshua Zahl

We establish an improved upper bound for the number of incidences between m points and n circles in three dimensions. The previous best known bound, originally established for the…

math.CO2012★ 32 cited

A Szemeredi-Trotter type theorem in $R^{4}$

Joshua Zahl

We show that $m$ points and $n$ two-dimensional algebraic surfaces in $R^{4}$ can have at most $O (m^{\frac{k}{2 k - 1}} n^{\frac{2 k - 2}{2 k - 1}} + m + n)$ incidences, provided that the…

math.CO2011★ 15 cited

An improved bound on the number of point-surface incidences in three dimensions

Joshua Zahl

We show that $m$ points and $n$ smooth algebraic surfaces of bounded degree in $R^{3}$ satisfying suitable nondegeneracy conditions can have at most $O(m^{\frac{2k}{3k-1}}n^…

math.CO2007★ 14 cited

On Universal Cycles for Multisets

Glenn Hurlbert, Tobias Johnson, Joshua Zahl

A Universal Cycle for t-multisets of [n]={1,...,n} is a cyclic sequence of $(t n + t - 1)$ integers from [n] with the property that each t-multiset of [n] appears exactly once c…

math.CO2006

Universal Cycles on 3-Multisets

Tobias L. Johnson, Joshua Zahl

Consider the collection of all t-multisets of {1,...,n}. A universal cycle on multisets is a string of numbers, each of which is between 1 and n, such that if these numbers are con…

On the number of pairwise touching cylinders in Rd

Improved Elekes-Szabó type estimates using proximity

On rich lenses in planar arrangements of circles and related problems

Distinct distances in the complex plane

Sphere tangencies, line incidences, and Lie's line-sphere correspondence

Counting higher order tangencies for plane curves

Cutting Algebraic Curves into Pseudo-segments and Applications

Curves in R4 and two-rich points

New bounds on curve tangencies and orthogonalities

A note on rich lines in truly high dimensional sets

Point-curve incidences in the complex plane

A semi-algebraic version of Zarankiewicz's problem

Few distinct distances implies no heavy lines or circles

Improved bounds for incidences between points and circles

A Szemeredi-Trotter type theorem in R4

An improved bound on the number of point-surface incidences in three dimensions

On Universal Cycles for Multisets

Universal Cycles on 3-Multisets

On the number of pairwise touching cylinders in $R^{d}$

Curves in $R^{4}$ and two-rich points

A Szemeredi-Trotter type theorem in $R^{4}$