Polynomial Partitioning Research Articles

Lp estimates of the maximal Schrödinger operator in [formula omitted]

We obtain Lp estimates of the maximal Schrödinger operator in Rn using polynomial partitioning, bilinear refined Strichartz estimates, and weighted restriction estimates.

Oscillatory integral operators with homogeneous phase functions

AbstractOscillatory integral operators with 1-homogeneous phase functions satisfying a convexity condition are considered. For these we show the Lp–Lp-estimates for the Fourier extension operator of the cone due to Ou–Wang via polynomial partitioning. For this purpose, we combine the arguments of Ou–Wang with the analysis of Guth–Hickman–Iliopoulou, who previously showed sharp Lp–Lp-estimates for non-homogeneous phase functions with variable coefficients under a convexity assumption. Furthermore, we provide examples exhibiting Kakeya compression, which shows a more restrictive range than dictated by the Knapp example in higher dimensions. We apply the oscillatory integral estimates to show new local smoothing estimates for wave equations on compact Riemannian manifolds (M, g) with dim M ≥ 3. This generalizes the argument for the Euclidean wave equation due to Gao–Liu–Miao–Xi.

Improved Bounds for the Expected Number of k-Sets

Given a finite set of points \(S\subset {\mathbb {R}}^d\), a k-set of S is a subset \(A\subset S\) of size k which can be strictly separated from \(S\setminus A\) by a hyperplane. Similarly, a k-facet of a point set S in general position is a subset \(\varDelta \subset S\) of size d such that the hyperplane spanned by \(\varDelta \) has k points from S on one side. For a probability distribution P on \({\mathbb {R}}^d\), we study \(E_P(k,n)\), the expected number of k-facets of a sample of n random points from P. When P is a distribution on \({\mathbb {R}}^2\) such that the measure of every line is 0, we show that \(E_P(k,n)=O(n(k+1)^{1/4})\). Our argument is based on a technique by Bárány and Steiger. We study how it may be possible to improve this bound using the continuous version of the polynomial partitioning theorem. This motivates a question concerning the points of intersection of an algebraic curve and the k-edge graph of a set of points. We also study a variation on the k-set problem for the set system whose set of ranges consists of all translations of some strictly convex body in the plane. The motivation is to show that the technique by Bárány and Steiger is tight for a natural family of set systems. For any such set system, we determine bounds for the expected number of k-sets which are tight up to logarithmic factors.

Conditional regression for single-index models

The single-index model is a statistical model for intrinsic regression where responses are assumed to depend on a single yet unknown linear combination of the predictors, allowing to express the regression function as E[Y|X]=f(⟨v,X⟩) for some unknown index vector v and link function f. Conditional methods provide a simple and effective approach to estimate v by averaging moments of X conditioned on Y, but depend on parameters whose optimal choice is unknown and do not provide generalization bounds on f. In this paper we propose a new conditional method converging at n rate under an explicit parameter characterization. Moreover, we prove that polynomial partitioning estimates achieve the 1-dimensional min-max rate for regression of Hölder functions when combined to any n-convergent index estimator. Overall this yields an estimator for dimension reduction and regression of single-index models that attains statistical optimality in quasilinear time.

Maximal subspace averages

We consider maximal operators associated to singular averages along finite subsets Σ of the Grassmannian Gr(d,n) of d-dimensional subspaces of Rn. The well studied d=1 case corresponds to the directional maximal function with respect to arbitrary finite subsets of Gr(1,n)=Sn−1. We provide a systematic study of all cases 1≤d<n and prove essentially sharp L2(Rn) bounds for the maximal subspace averaging operator in terms of the cardinality of Σ, with no assumption on the structure of Σ. In the codimension 1 case, that is n=d+1, we prove the precise critical weak (2,2)-bound.Drawing on the analogy between maximal subspace averages and (d,n)-Nikodym maximal averages, we also formulate the appropriate maximal Nikodym conjecture for general 1<d<n by providing examples that determine the critical Lp-space for the (d,n)-Nikodym problem. Unlike the d=1 case, the maximal Kakeya and Nikodym problems are shown not to be equivalent when d>1. In this context, we prove the best possible L2(Rn)-bound for the (d,n)-Nikodym maximal function for all combinations of dimension and codimension.Our estimates rely on Fourier analytic almost orthogonality principles, combined with polynomial partitioning, but we also use spatial analysis based on the precise calculation of intersections of d-dimensional plates in Rn.

Local Extension Estimates for the Hyperbolic Hyperboloid in Three Dimensions

We establish Fourier extension estimates for compact subsets of the hyperbolic hyperboloid in three dimensions via polynomial partitioning.

Subquadratic algorithms for some 3Sum-hard geometric problems in the algebraic decision-tree model

We present subquadratic algorithms in the algebraic decision-tree model for several 3Sum-hard geometric problems, all of which can be reduced to the following question: Given two sets A, B, each consisting of n pairwise disjoint segments in the plane, and a set C of n triangles in the plane, we want to count, for each triangle Δ∈C, the number of intersection points between the segments of A and those of B that lie in Δ. We present solutions in the algebraic decision-tree model whose cost is O(n60/31+ε), for any ε>0.Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal et al. (2021) [3]. A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the order type of the lines, a “handicap” that turns out to be beneficial for speeding up our algorithm.

Fourier restriction for smooth hyperbolic 2-surfaces

We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in mathbb {R}^3. Our approach exploits in a crucial way the underlying hyperbolic geometry, which leads to a novel notion of strong transversality and corresponding “exceptional” sets. For the division of these exceptional sets we make crucial and perhaps surprising use of a lemma on level sets for sufficiently smooth one-variate functions from a previous article of ours.

On Ray Shooting for Triangles in 3-Space and Related Problems

We consider several problems that involve lines in three dimensions, and present improved algorithms for solving them. The problems include (i) ray shooting amid triangles in $R^3$, (ii) reporting intersections between query lines (segments, or rays) and input triangles, as well as approximately counting the number of such intersections, (iii) computing the intersection of two nonconvex polyhedra, (iv) detecting, counting, or reporting intersections in a set of lines in $R^3$, and (v) output-sensitive construction of an arrangement of triangles in three dimensions. Our approach is based on the polynomial partitioning technique. For example, our ray-shooting algorithm processes a set of $n$ triangles in $R^3$ into a data structure for answering ray shooting queries amid the given triangles, which uses $O(n^{3/2+\varepsilon})$ storage and preprocessing, and answers a query in $O(n^{1/2+\varepsilon})$ time, for any $\varepsilon>0$. This is a significant improvement over known results, obtained more than 25 years ago, in which, with this amount of storage, the query time bound is roughly $n^{5/8}$. The algorithms for the other problems have similar performance bounds, with similar improvements over previous results. We also derive a nontrivial improved tradeoff between storage and query time. Using it, we obtain algorithms that answer $m$ queries on $n$ objects in \[ \max \left\{ O(m^{2/3}n^{5/6+\varepsilon} + n^{1+\varepsilon}),\; O(m^{5/6+\varepsilon}n^{2/3} + m^{1+\varepsilon}) \right\} \] time, for any $\varepsilon>0$, again an improvement over the earlier bounds.

L2 bounds for a maximal directional Hilbert transform

Given any finite direction set $\Omega$ of cardinality $N$ in Euclidean space, we consider the maximal directional Hilbert transform $H_{\Omega}$ associated to this direction set. Our main result provides an essentially sharp uniform bound, depending only on $N$, for the $L^2$ operator norm of $H_{\Omega}$ in dimensions 3 and higher. The main ingredients of the proof consist of polynomial partitioning tools from incidence geometry and an almost-orthogonality principle for $H_{\Omega}$. The latter principle can also be used to analyze special direction sets $\Omega$, and derive sharp $L^2$ estimates for the corresponding operator $H_{\Omega}$ that are typically stronger than the uniform $L^2$ bound mentioned above. A number of such examples are discussed.

A restriction estimate in R3 using brooms

If f is a function supported on the truncated paraboloid in R3 and E is the corresponding extension operator, then we prove that for all p>3+3∕13, ‖Ef‖Lp(R3)≤C‖f‖L∞. The proof combines Wolff’s two ends argument with polynomial partitioning techniques. We also observe some geometric structures in the wave packets.

A Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning

AbstractWe consider a surface with negative curvature in $${{\mathbb {R}}}^3,$$ R 3 , which is a cubic perturbation of the saddle. For this surface, we prove a new restriction theorem, analogous to the theorem for paraboloids proved by L. Guth in 2016. This specific perturbation has turned out to be of fundamental importance also to the understanding of more general classes of one-variate perturbations, and we hope that the present paper will further help to pave the way for the study of general perturbations of the saddle by means of the polynomial partitioning method.

Incidences between points and curves with almost two degrees of freedom

We study incidences between points and (constant-degree algebraic) curves in three dimensions, taken from a family C of curves that have almost two degrees of freedom, meaning that (i) every pair of curves of C intersect in O(1) points, (ii) for any pair of points p, q, there are only O(1) curves of C that pass through both points, and (iii) there exists a 6-variate real polynomial F of constant degree, so that a pair p, q of points admit a curve of C that passes through both of them if and only if F(p,q)=0. (As an example, the family of unit circles in R3 that pass through some fixed point is such a family.)We begin by studying two specific instances of this scenario. The first instance deals with the case of unit circles in R3 that pass through some fixed point (so called anchored unit circles). In the second case we consider tangencies between directed points and circles in the plane, where a directed point is a pair (p,u), where p is a point in the plane and u is a direction, and (p,u) is tangent to a circle γ if p∈γ and u is the direction of the tangent to γ at p. A lifting transformation due to Ellenberg et al. [3] maps these tangencies to incidences between points and curves (‘lifted circles’) in three dimensions. In both instances we have a family of curves in R3 with almost two degrees of freedom.We show that the number of incidences between m points and n anchored unit circles in R3, as well as the number of tangencies between m directed points and n arbitrary circles in the plane, is O(m3/5n3/5+m+n).We then derive a similar incidence bound, with a few additional terms, for more general families C of curves in R3 with almost two degrees of freedom, under a few additional natural assumptions.The proofs follow standard techniques, based on polynomial partitioning, but they face a critical novel issue involving the analysis of surfaces that are infinitely ruled by the respective family of curves, as well as of surfaces in a dual three-dimensional space that are infinitely ruled by the respective family of suitably defined dual curves. We either show that no such surfaces exist, or develop and adapt techniques for handling incidences on such surfaces.The general bound that we obtain isI(P,C)=O(m3/5n3/5+(m11/15n2/5+n8/9)δ1/3+m2/3n1/3π1/3+m+n), where π (resp., δ) is the maximal number of curves (resp., dual curves) that lie on a common surface that is infinitely ruled by the family of these curves.

A cone restriction estimate using polynomial partitioning

We obtain an improved Fourier restriction estimate for a truncated cone using the method of polynomial partitioning in dimension n\geq 3 , which in particular solves the cone restriction conjecture for n=5 , and recovers the sharp range for 3\leq n\leq 4 . The main ingredient of the proof is a k -broad estimate for the cone extension operator, which is a weak version of the k -linear cone restriction conjecture for 2\leq k\leq n .

Restriction estimates to complex hypersurfaces

The restriction problem is relatively well understood for hypersurfaces and recent progresses have been made by means of bilinear and multilinear approaches and, more recently, by means of polynomial partitioning combined with those approaches. However, for surfaces with codimension bigger than 1, bilinear and multilinear generalizations of restriction estimates are more involved and effectiveness of these multilinear estimates is not so well understood yet. Regarding the restriction problem for surfaces with codimensions bigger than 1, the current state of the art is still at the level of the TT⁎ method which is known to be useful for obtaining Lq–L2 estimates. In this paper, we consider a special type of codimension 2 surfaces, given by the graphs of complex analytic functions, and attempt to make progress beyond the L2 restriction estimates.

Lp-estimate of Schrödinger maximal function in higher dimensions

Almost everywhere convergence on the solution of Schrödinger equation is an important problem raised by Carleson in harmonic analysis. In recent years, this problem was essentially solved by building the sharp Lp-estimate of Schrödinger maximal function. Du-Guth-Li in [8] proved the sharp Lp-estimates for all p≥2 in R2+1. Du-Zhang in [12] proved the sharp L2-estimate in Rn+1 with n≥3, but for p>2 the sharp Lp-estimate of Schrödinger maximal function is still unknown. In this paper, we obtain partial results on this problem by using polynomial partitioning and refined Strichartz estimates.

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.

Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications

In 2015, Guth proved that if $\EuScript{S}$ is a collection of $n$ $g$-dimensional semialgebraic sets in ${\mathbb{R}}^d$ and if $D\geq 1$ is an integer, then there is a $d$-variate polynomial $P$ ...

Maximal directional operators along algebraic varieties

We establish the sharp growth order, up to epsilon losses, of the $L^2$-norm of the maximal directional averaging operator along a finite subset $V$ of a polynomial variety of arbitrary dimension $m$, in terms of cardinality. This is an extension of the works by C\'ordoba, for one-dimensional manifolds, Katz for the circle in two dimensions, and Demeter for the 2-sphere. For the case of directions on the two-dimensional sphere we improve by a factor of $\sqrt{\log N}$ on the best known bound, due to Demeter, and we obtain a sharp estimate for our model operator. Our results imply new $L^2$-estimates for Kakeya type maximal functions with tubes pointing along polynomial directions. Our proof technique is novel and in particular incorporates an iterated scheme of polynomial partitioning on varieties adapted to directional operators, in the vein of Guth, Guth-Katz, and Zahl.

Eliminating Depth Cycles Among Triangles in Three Dimensions

The vertical depth relation among n pairwise openly disjoint triangles in 3-space may contain cycles. We show that, for any \(\varepsilon >0\), the triangles can be cut into \(O(n^{3/2+\varepsilon })\) connected semialgebraic pieces, whose description complexity depends only on the choice of \(\varepsilon \), such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case. The pieces can be constructed efficiently. This work extends the recent study by two of the authors (Discrete Comput. Geom. 59(3), 725–741 (2018)) on eliminating depth cycles among lines in 3-space. Our approach is again algebraic, and makes use of a recent variant of the polynomial partitioning technique, due to Guth (Math. Proc. Camb. Philos. Soc. 159(3), 459–469 (2015)), which leads to a recursive algorithm for cutting the triangles. In contrast to the case of lines, our analysis here is considerably more involved, due to the two-dimensional nature of the objects being cut, so additional tools, from topology and algebra, need to be brought to bear. Our result makes significant progress towards resolving a decades-old open problem in computational geometry, motivated by hidden-surface removal in computer graphics. In addition, we generalize our bound to well-behaved patches of two-dimensional algebraic surfaces of constant degree.