Counting joints with multiplicities

Let $\mathfrak{L}$ be a collection of $L$ lines in $\R^3$ and $J$ the set of joints formed by $\mathfrak{L}$, i.e. the set of points each of which lies in at least 3 non-coplanar lines of $\mathfrak{L}$. It is known that $|J| \lesssim L^{3/2}$ (first proved by Guth and Katz). For each joint $x \in J$, let the multiplicity $N(x)$ of $x$ be the number of triples of non-coplanar lines through $x$. We prove here that $\sum_{x \in J}N(x)^{1/2} \lesssim L^{3/2}$, while in the last section we extend this result to real algebraic curves of uniformly bounded degree in $\R^3$, as well as to curves in $\R^3$ parametrised by real polynomials of uniformly bounded degree.Comment: More details in section 4. Typos corrected. The main results are unchange

Counting joints in vector spaces over arbitrary fields

We give a proof of the "folklore" theorem that the Kaplan--Sharir--Shustin/Quilodr\'an result on counting joints associated to a family of lines holds in vector spaces over arbitrary fields, not just the reals. We also discuss a distributional estimate on the multiplicities of the joints in the case that the family of lines is sufficiently generic.Comment: Not intended for publication. References added and other minor edits in this versio

Counting multijoints

Let $\mathfrak{L}_1$, $\mathfrak{L}_2$, $\mathfrak{L}_3$ be finite collections of $L_1$, $L_2$, $L_3$, respectively, lines in $\mathbb{R}^3$, and $J(\mathfrak{L}_1, \mathfrak{L}_2,\mathfrak{L}_3)$ the set of multijoints formed by them, i.e. the set of points $x \in \mathbb{R}^3$, each of which lies in at least one line $l_i \in \mathfrak{L}_i$, for all $i=1,2,3$, such that the directions of $l_1$, $l_2$ and $l_3$ span $\mathbb{R}^3$. We prove here that $|J(\mathfrak{L}_1, \mathfrak{L}_2,\mathfrak{L}_3)|\lesssim (L_1L_2L_3)^{1/2}$, and we extend our results to multijoints formed by real algebraic curves in $\mathbb{R}^3$ of uniformly bounded degree, as well as by curves in $\mathbb{R}^3$ parametrised by real univariate polynomials of uniformly bounded degree. The multijoints problem is a variant of the joints problem, as well as a discrete analogue of the endpoint multilinear Kakeya problem.Comment: 25 pages. arXiv admin note: substantial text overlap with arXiv:1312.543

Discrete analogues of Kakeya problems

This thesis investigates two problems that are discrete analogues of two harmonic analytic problems which lie in the heart of research in the field. More specifically, we consider discrete analogues of the maximal Kakeya operator conjecture and of the recently solved endpoint multilinear Kakeya problem, by effectively shrinking the tubes involved in these problems to lines, thus giving rise to the problems of counting joints and multijoints with multiplicities. In fact, we effectively show that, in R3, what we expect to hold due to the maximal Kakeya operator conjecture, as well as what we know in the continuous case due to the endpoint multilinear Kakeya theorem by Guth, still hold in the discrete case. In particular, let L be a collection of L lines in R3 and J the set of joints formed by L, that is, the set of points each of which lies in at least three non-coplanar lines of L. It is known that |J| = O(L3/2) ( first proved by Guth and Katz). For each joint x ∈ J, let the multiplicity N(x) of x be the number of triples of non-coplanar lines through x. We prove here that X x2J N(x)1=2 = O(L3=2); while we also extend this result to real algebraic curves in R3 of uniformly bounded degree, as well as to curves in R3 parametrized by real univariate polynomials of uniformly bounded degree. The multijoints problem is a variant of the joints problem, involving three finite collections of lines in R3; a multijoint formed by them is a point that lies in (at least) three non-coplanar lines, one from each collection. We finally present some results regarding the joints problem in different field settings and higher dimensions

Incidence bounds on multijoints and generic joints

A point $x \in \mathbb{F}^n$ is a joint formed by a finite collection $\mathfrak{L}$ of lines in $\mathbb{F}^n$ if there exist at least $n$ lines in $\mathfrak{L}$ through $x$ that span $\mathbb{F}^n$. It is known that there are $\lesssim_n |\mathfrak{L}|^{\frac{n}{n-1}}$ joints formed by $\mathfrak{L}$. We say that a point $x \in \mathbb{F}^n$ is a multijoint formed by the finite collections $\mathfrak{L}_1,\ldots,\mathfrak{L}_n$ of lines in $\mathbb{F}^n$ if there exist at least $n$ lines through $x$, one from each collection, spanning $\mathbb{F}^n$. We show that there are $\lesssim_n (|\mathfrak{L}_1|\cdots |\mathfrak{L}_n|)^{\frac{1}{n-1}}$ such points for any field $\mathbb{F}$ and $n=3$, as well as for $\mathbb{F}=\mathbb{R}$ and any $n \geq 3$. Moreover, we say that a point $x \in \mathbb{F}^n$ is a generic joint formed by a finite collection $\mathfrak{L}$ of lines in $\mathbb{F}^n$ if each $n$ lines of $\mathfrak{L}$ through $x$ form a joint there. We show that, for $\mathbb{F}=\mathbb{R}$ and any $n \geq 3$, there are $\lesssim_n \frac{|\mathfrak{L}|^{\frac{n}{n-1}}}{k^{\frac{n+1}{n-1}}}+\frac{|\mathfrak{L}|}{k}$ generic joints formed by $\mathfrak{L}$, each lying in $\sim k$ lines of $\mathfrak{L}$. This result generalises, to all dimensions, a (very small) part of the main point-line incidence theorem in $\mathbb{R}^3$ in \cite{Guth_Katz_2010} by Guth and Katz. Finally, we generalise our results in $\mathbb{R}^n$ to the case of multijoints and generic joints formed by real algebraic curves.Comment: Some errors corrected. Theorem 4.4 is now slightly stronger than its previous version. To appear in Discrete Comput. Geo

A multilinear Fourier extension identity on ℝn

We prove an elementary multilinear identity for the Fourier extension operator on $\mathbb{R}^n$, generalising to higher dimensions the classical bilinear extension identity in the plane. In the particular case of the extension operator associated with the paraboloid, this provides a higher dimensional extension of a well-known identity of Ozawa and Tsutsumi for solutions to the free time-dependent Schr\"odinger equation. We conclude with a similar treatment of more general oscillatory integral operators whose phase functions collectively satisfy a natural multilinear transversality condition. The perspective we present has its origins in work of Drury.Comment: To appear in Mathematical Research Letter

Joints formed by lines and a k-plane, and a discrete estimate of Kakeya type

Let $\mathcal{L}$ be a family of lines and let $\mathcal{P}$ be a family of $k$-planes in $\mathbb{F}$$^n$ where $\mathbb{F}$ is a field. In our first result we show that the number of joints formed by a $k$-plane in $\mathcal{P}$ together with ${(n-k)}$ lines in $\mathcal{L}$ is $\mathcal{O}$$_n$(|$\mathcal{L}$||$\mathcal{P}$|$^{1/(n-k)}$. This is the first sharp result for joints involving higher-dimensional affine subspaces, and it holds in the setting of arbitrary fields $\mathbb{F}$. In contrast, for our second result, we work in the three-dimensional Euclidean space $\mathbb{R}$$^3$, and we establish the Kakeya-type estimate $$\sum_{x \in J} \left(\sum_{\ell \in \mathcal{L}} \chi_\ell(x)\right)^{3/2} \lesssim |\mathcal{L}|^{3/2}$$ where $J$ is the set of joints formed by $\mathcal{L}$; such an estimate fails in the setting of arbitrary fields. This result strengthens the known estimates for joints, including those counting multiplicities. Additionally, our techniques yield significant structural information on quasi-extremisers for this inequality

Flow monotonicity and Strichartz inequalities

We identify complete monotonicity properties underlying a variety of well-known sharp Strichartz inequalities in euclidean space