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Larry Guth

Publications and source records attributed to Larry Guth.

At least 37 records · Page 2Linked to original sources
arXiv2023

Restriction estimates for quadratic manifolds of arbitrary codimensions

The restriction conjecture is one of the famous problems in harmonic analysis. There have been many methods developed in the study of the conjecture for the paraboloid. In this paper, we generalize the multilinear method of Bourgain and Guth for the paraboloid, and obtain restriction estimates for all quadratic manifolds of arbitrary codimensions. In particular, our theorem recovers the main theorem of Bourgain and Guth for the paraboloid. A new ingredient is a covering lemma for varieties whose proof relies on Tarski's projection theorem in real algebraic geometry. We also provide algorithms to compute several algebraic quantities that naturally appear in the argument. These algorithms rely on a cylindrical decomposition in real algebraic geometry.

math.CA↗
arXiv2022

An exceptional set estimate for restricted projections to lines in $\mathbb{R}^3$

Let $γ:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det\big(γ(θ),γ'(θ),γ''(θ)\big)\neq 0$. For each $θ\in[0,1]$, let $l_θ=\{tγ(θ):t\in\mathbb{R}\}$ and $ρ_θ:\mathbb{R}^3\rightarrow l_θ$ be the orthogonal projections. We prove an exceptional set estimate. For any Borel set $A\subset\mathbb{R}^3$ and $0\le s\le 1$, define $E_s(A):=\{θ\in[0,1]: \text{dim}(ρ_θ(A))<s\}$. We have $\text{dim}(E_s(A))\le 1+\frac{s-\text{dim}(A)}{2}$.

math.CA↗
arXiv2022

Amplitude dependent wave envelope estimates for the cone in $\mathbb{R}^3$

For functions $f$ with Fourier transform supported in the truncated cone, we bound superlevel sets $\{x\in\mathbb{R}^3:|f(x)|>α\}$ using an $α$-dependent version of the wave envelope estimate of Guth--Wang--Zhang. Our estimates imply both sharp square function and decoupling inequalities for the cone. We also obtain sharp small cap decoupling for the cone, where small caps $γ$ subdivide canonical $1\times R^{-1/2}\times R^{-1}$ planks into $R^{-β_2}\times R^{-β_1}\times R^{-1}$ sub-planks, for $β_1\in[\frac{1}{2},1]$ and $β_2\in[0,1]$.

math.CA↗
arXiv2022

Small cap decoupling for the moment curve in $\mathbb{R}^3$

We prove sharp small cap decoupling estimates for the moment curve in $\mathbb{R}^3$. Our formulation of the small caps is motivated by a conjecture about $L^p$ estimates for exponential sums from the small cap decoupling paper of Demeter, Guth, and Wang.

math.CA↗
arXiv2022

Decoupling estimates in Fourier analysis

Decoupling is a recent development in Fourier analysis, which has applications in harmonic analysis, PDE, and number theory. We survey some applications of decoupling and some of the ideas in the proof. This survey is aimed at a general mathematical audience. It is based on my 2022 ICM talk.

math.CA↗
arXiv2021

Macroscopic scalar curvature and codimension 2 width

We show that a complete $3$-dimensional Riemannian manifold $M$ with finitely generated first homology has macroscopic dimension $1$ if it satisfies the following "macroscopic curvature" assumptions: every ball of radius $10$ in $M$ has volume at most $4$, and every loop in every ball of radius $1$ in $M$ is null-homologous in the concentric ball of radius $2$.

math.DG↗
arXiv2021

Sharp Szemerédi-Trotter Constructions in the Plane

We present a new family of sharp examples for the Szemerédi-Trotter theorem. These are the first examples not based on a rectangular lattice. We also include an application to the discrete inverse Loomis-Whitney problem.

math.CO↗
arXiv2021

Sharp superlevel set estimates for small cap decouplings of the parabola

We prove sharp bounds for the size of superlevel sets $\{x\in \mathbb{R}^2:|f(x)|>α\}$ where $α>0$ and $f:\mathbb{R}^2\to\mathbb{C}$ is a Schwartz function with Fourier transform supported in an $R^{-1}$-neighborhood of the truncated parabola $\mathbb{P}^1$. These estimates imply the small cap decoupling theorem for $\mathbb{P}^1$ of Demeter, Guth, and Wang, and the canonical decoupling theorem for $\mathbb{P}^1$ of Bourgain and Demeter. New $(\ell^q,L^p)$ small cap decoupling inequalities also follow from our sharp level set estimates.

math.CA↗
arXiv2021

Decoupling inequalities for short generalized Dirichlet sequences

We study decoupling theory for functions on $\mathbb{R}$ with Fourier transform supported in a neighborhood of short Dirichlet sequences $\{\log n\}_{n=N+1}^{N+N^{1/2}}$, as well as sequences with similar convexity properties. We utilize the wave packet structure of functions with frequency support near an arithmetic progression.

math.CA↗
arXiv2020

Three Applications of Entropy to Gerrymandering

This preprint is an exploration in how a single mathematical idea - entropy - can be applied to redistricting in a number of ways. It's meant to be read not so much as a call to action for entropy, but as a case study illustrating one of the many ways math can inform our thinking on redistricting problems. This preprint was prepared as a chapter in the forthcoming edited volume Political Geometry, an interdisciplinary collection of essays on redistricting. (mggg.org/gerrybook)

cs.IT↗
arXiv2020

Improved decoupling for the parabola

We prove an $(l^2, l^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. In the appendix, we present an application to the six-order correlation of the integer solutions to $x^2+y^2=m$.

math.CA↗
arXiv2020

Small cap decouplings

We develop a toolbox for proving decouplings into boxes with diameter smaller than the canonical scale. As an application of this new technique, we solve three problems for which earlier methods have failed. We start by verifying the small cap decoupling for the parabola. Then we find sharp estimates for exponential sums with small frequency separation on the moment curve in $\mathbb{R}^3$. This part of the work relies on recent improved Kakeya-type estimates for planar tubes, as well as on new multilinear incidence bounds for plates and planks. We also combine our method with the recent advance on the reverse square function estimate, in order to prove small cap decoupling into square-like caps for the two dimensional cone. The Appendix by Roger Heath-Brown contains an application of the new exponential sum estimates for the moment curve, to the Riemann zeta-function.

math.CA↗
arXiv2019

On the discretized sum-product problem

We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if $A\subset\mathbb{R}$ is a $(δ,1/2)_1$-set in the sense of Katz and Tao, then either $A+A$ or $A.A$ must have measure at least $|A|^{1-\frac{1}{68}}$

math.CA↗
arXiv2019

Sharp estimates for oscillatory integral operators via polynomial partitioning

The sharp range of $L^p$-estimates for the class of Hörmander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments.

math.CA↗
arXiv2019

Polynomial Wolff axioms and Kakeya-type estimates in $\mathbb{R}^4$

We establish new linear and trilinear bounds for collections of tubes in $\mathbb{R}^4$ that satisfy the polynomial Wolff axioms. In brief, a collection of $δ$-tubes satisfies the Wolff axioms if not too many tubes can be contained in the $δ$-neighborhood of a plane. A collection of tubes satisfies the polynomial Wolff axioms if not too many tubes can be contained in the $δ$-neighborhood of a low degree algebraic variety. First, we prove that if a set of $δ^{-3}$ tubes in $\mathbb{R}^4$ satisfies the polynomial Wolff axioms, then the union of the tubes must have volume at least $δ^{1-1/40}$. We also prove a more technical statement which is analogous to a maximal function estimate at dimension $3+1/40$. Second, we prove that if a collection of $δ^{-3}$ tubes in $\mathbb{R}^4$ satisfies the polynomial Wolff axioms, and if most triples of intersecting tubes point in three linearly independent directions, then the union of the tubes must have volume at least $δ^{3/4}$. Again, we also prove a slightly more technical statement which is analogous to a maximal function estimate at dimension $3+1/4$. We conjecture that every Kakeya set satisfies the polynomial Wolff axioms, but we are unable to prove this. If our conjecture is correct, it implies a Kakeya maximal function estimate at dimension $3+1/40$, and in particular this implies that every Kakeya set in $\mathbb{R}^4$ must have Hausdorff dimension at least $3+1/40$. This would be an improvement over the current best bound of 3, which was established by Wolff in 1995.

math.CA↗