Incidence estimates for well spaced tubes
We prove analogues of the Szemerédi-Trotter theorem and other incidence theorems using $δ$-tubes in place of straight lines, assuming that the $δ$-tubes are well-spaced in a strong sense.
math.CA↗
arXiv subjects
Larry Guth
Publications and source records attributed to Larry Guth.
On Falconer's distance set problem in the plane
If $E \subset \mathbb{R}^2$ is a compact set of Hausdorff dimension greater than $5/4$, we prove that there is a point $x \in E$ so that the set of distances $\{ |x-y| \}_{y \in E}$ has positive Lebesgue measure.
math.CA↗
Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimates
We obtain partial improvement toward the pointwise convergence problem of Schrödinger solutions, in the general setting of fractal measure. In particular, we show that, for $n\geq 3$, $\lim_{t \to 0} e^{itΔ}f(x) = f(x)$ almost everywhere with respect to Lebesgue measure for all $f \in H^s (\mathbb{R}^n)$ provided that $s>(n+1)/2(n+2)$. The proof uses linear refined Strichartz estimates. We also prove a multilinear refined Strichartz using decoupling and multilinear Kakeya.
math.CA↗
Weighted restriction estimates and application to Falconer distance set problem
We prove some weighted Fourier restriction estimates using polynomial partitioning and refined Strichartz estimates. As application we obtain improved spherical average decay rates of the Fourier transform of fractal measures, and therefore improve the results for the Falconer distance set conjecture in three and higher dimensions.
math.CA↗
Curves in $\mathbb{R}^4$ and two-rich points
We obtain a new bound on the number of two-rich points spanned by an arrangement of low degree algebraic curves in $\mathbb{R}^4$. Specifically, we show that an arrangement of $n$ algebraic curves determines at most $C_εn^{4/3+3ε}$ two-rich points, provided at most $n^{2/3+2ε}$ curves lie in any low degree hypersurface and at most $n^{1/3+ε}$ curves lie in any low degree surface. This result follows from a structure theorem about arrangements of curves that determine many two-rich points.
math.CO↗
Restriction estimates using polynomial partitioning II
We improve the estimates in the restriction problem in dimension $n \ge 4$. To do so, we establish a weak version of a $k$-linear restriction estimate for any $k$. The exponents in this weak $k$-linear estimate are sharp for all $k$ and $n$.
math.CA↗
2-complexes with large 2-girth
The 2-girth of a 2-dimensional simplicial complex $X$ is the minimum size of a non-zero 2-cycle in $H_2(X, \mathbb{Z}/2)$. We consider the maximum possible girth of a complex with $n$ vertices and $m$ 2-faces. If $m = n^{2 + α}$ for $α< 1/2$, then we show that the 2-girth is at most $4 n^{2 - 2 α}$ and we prove the existence of complexes with 2-girth at least $c_{α, ε} n^{2 - 2 α- ε}$. On the other hand, if $α> 1/2$, the 2-girth is at most $C_α$. So there is a phase transition as $α$ passes 1/2. Our results depend on a new upper bound for the number of combinatorial types of triangulated surfaces with $v$ vertices and $f$ faces.
math.AT↗
A sharp Schrodinger maximal estimate in $\mathbb{R}^2$
We show that $\lim_{t \to 0} e^{itΔ}f(x) = f(x)$ almost everywhere for all $f \in H^s (\mathbb{R}^2)$ provided that $s>1/3$. This result is sharp up to the endpoint. The proof uses polynomial partitioning and decoupling.
math.CA↗
Strichartz estimates for the Schrodinger equation on irrational tori
We prove Strichartz estimates over large time scales for the Schrodinger equation set on irrational tori. They are optimal for Lebesgue exponents $p > 6$.
math.AP↗
Ruled surface theory and incidence geometry
This is an expository paper about applications of ruled surface theory in incidence geometry. It surveys the results that have been proven, gives an overview of the methods, and discusses some open problems and further directions. It will appear in the book Journey Through Discrete Mathematics, a Tribute to Jiri Matousek, edited by Martin Loebl, Jaroslav Nesetril and Robin Thomas, due to be published by Springer.
math.CO↗
Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three
We prove the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three. This will be a consequence of a sharp decoupling inequality for curves
math.NT↗
Algebraic curves, rich points, and doubly-ruled surfaces
We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with $n<\!\!<(\operatorname{char}(k))^2$ or $\operatorname{char}(k)=0$. Then either A) there are at most $O(n^{3/2})$ points in $k^3$ hit by at least two curves, or B) at least $Ω(n^{1/2})$ curves from $\mathcal{L}$ must lie on a bounded-degree surface, and many of the curves must form two "rulings" of this surface. We also develop several new tools including a generalization of the classical flecnode polynomial of Salmon and new algebraic techniques for dealing with this generalized flecnode polynomial.
math.AG↗
Polynomial partitioning for a set of varieties
Given a set $Γ$ of low-degree k-dimensional varieties in $\mathbb{R}^n$, we prove that for any $D \ge 1$, there is a non-zero polynomial $P$ of degree at most $D$ so that each component of $\mathbb{R}^n \setminus Z(P)$ intersects $O(D^{k-n} |Γ|)$ varieties of $Γ$.
math.AG↗
Volumes of balls in Riemannian manifolds and Uryson width
If $(M^n, g)$ is a closed Riemannian manifold where every unit ball has volume at most $ε_n$ (a sufficiently small constant), then the $(n-1)$-dimensional Uryson width of $(M^n, g)$ is at most 1.
math.DG↗
Amenable groups and smooth topology of 4-manifolds
A smooth five-dimensional s-cobordism becomes a smooth product if stabilized by a finite number n of $S^2xS^2x[0,1]$'s. We show that for amenable fundamental groups, the minimal n is subextensive in covers, i.e., n(cover)/index(cover) has limit 0. We focus on the notion of sweepout width, which is a bridge between 4-dimensional topology and coarse geometry.
math.GT↗
A restriction estimate using polynomial partitioning
If $S$ is a smooth compact surface in $\mathbb{R}^3$ with strictly positive second fundamental form, and $E_S$ is the corresponding extension operator, then we prove that for all $p > 3.25$, $\| E_S f\|_{L^p(\mathbb{R}^3)} \le C(p,S) \| f \|_{L^\infty(S)}$. The proof uses polynomial partitioning arguments from incidence geometry.
math.CA↗
Distinct distance estimates and low degree polynomial partitioning
We give a shorter proof of a slightly weaker version of a theorem of Nets Katz and the author. We prove that if a set of $L$ lines in $\mathbb{R}^3$ contains at most $L^{1/2}$ lines in any low degree algebraic surface, then the number of $r$-rich points is at most $C_εL^{(3/2) + ε} r^{-2}$. Nets and I used this estimate to prove a distinct distance estimate for points in the plane. With the slightly weaker theorem in this paper, we get a slightly weaker distinct distance estimate: any set of $N$ points in $\mathbb{R}^2$ determines at least $c_εN^{1 - ε}$ distinct distances.
math.CO↗
A short proof of the multilinear Kakeya inequality
We give a short proof of a slightly weaker version of the multilinear Kakeya inequality proven by Bennett, Carbery, and Tao.
math.AP↗