The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and are related to many problems in harmonic analysis, PDE, and number theory. In this paper we initiate the study of these problems on finite fields. The restriction problem then becomes a question about general bounds for certain types of exponential sums, while the Kakeya problem has an algebraic geometry flavor, asking for the extent to which lines in different directions can overlap. In many cases the Euclidean arguments carry over easily to the finite setting (and are in fact somewhat cleaner), but there are some new phenomena in the finite case which deserve closer study.
In the finite field setting, we show that the restriction conjecture associated to any one of a large family of $d=2n+1$ dimensional quadratic surfaces implies the $n+1$ dimensional Kakeya conjecture (Dvir's theorem). This includes the case of the paraboloid over finite fields in which $-1$ \emph{is} a square. We are able to partially reverse this implication using the sharp Kakeya maximal operator estimates of Ellenberg, Oberlin and Tao to establish the first finite field restriction estimates beyond the Stein-Tomas exponent in this setting.
The Fourier restriction phenomenon and the size of Kakeya sets are explored in the setting of the ring of integers modulo $N$ for general $N$ and a striking similarity with the corresponding euclidean problems is observed. One should contrast this with known results in the finite field setting.
We show that a certain conjectured optimal reverse Littlewood- Paley inequality would, if true, imply sharp results for the Kakeya maximal function, the Bochner-Riesz means and the Fourier restriction operator.
Bilinear restriction estimates have been appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimates (together with the analogues for Kakeya estimates). As a consequence we improve the $(L^p,L^p)$ spherical restriction theorem of Wolff from $p > 42/11$ to $p > 34/9$, and also obtain a sharp $(L^p,L^q)$ spherical restriction theorem for $q > 4 - 5/27$.
We prove $d$-linear analogues of the classical restriction and Kakeya conjectures in $\R^d$. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variable-coefficient problems and the so-called "joints" problem, as well as presenting some $n$-linear analogues for $n<d$.
We revisit the multilinear Kakeya, curved Kakeya, restriction, and oscillatory integral estimates that were obtained in paper of Bennett, Carbery, and the author using a heat flow monotonicity method applied to a fractional Cartesian product, together with induction on scales arguments. Many of these estimates contained losses of the form $R^\varepsilon$ (or $\log^{O(1)} R$) for some scale factor $R$. By further developing the heat flow method, and applying it directly for the first time to the multilinear curved Kakeya and restriction settings, we are able to eliminate these losses, as long a…
In a recent paper [Ann. of Math. 189 (2019), 837--861], Du and Zhang proved a fractal Fourier restriction estimate and used it to establish the sharp $L^2$ estimate on the Schrödinger maximal function in $\Bbb R^n$, $n \geq 2$. In this paper, we show that the Du-Zhang estimate is the endpoint of a family of fractal restriction estimates such that each member of the family (other than the original) implies a sharp Kakeya result in $\Bbb R^n$ that is closely related to the polynomial Wolff axioms. We also prove that all the estimates of our family are true in $\Bbb R^2$.
Here we show some results related with Kakeya conjecture which says that for any integer $n\geq 2$, a set containing line segments in every dimension in $\mathbb{R}^n$ has full Hausdorff dimension as well as box dimension. We proved here that the Kakeya books, which are Kakeya sets with some restrictions on positions of line segments have full box dimension. We also prove here a relation between the projection property of Kakeya sets and the Kakeya conjecture. If for any Kakeya set $K\subset\mathbb{R}^n$, the Hausdorff dimension of orthogonal projections on $k\leq n$ subspaces is independent o…
We investigate analogues for curves of the Kakeya problem for straight lines. These arise from H"ormander-type oscillatory integrals in the same way as the straight line case comes from the restriction and Bochner-Riesz problems. Using some of the geometric and arithmetic techniques developed for the straight line case by Bourgain, Wolff, Katz and Tao, we are able to prove positive results for families of parabolas whose coefficients satisfy certain algebraic conditions.
We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To take advantage of this, we prove that direction-separated tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether, this yields improved bounds for the Kakeya maximal conjecture in $\mathbb{R}^n$ with $n=5$ or $n\ge 7$ and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions.
We improve the $L^{p}\rightarrow L^p$ restriction estimate in $\mathbb{R}^3$ to the range $p>3+3/14$, based on some Kakeya type incidence estimates and the refined decoupling theorem.
We consider Guth's approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions. Consequences for the Kakeya conjecture are also considered.
In 1901, Severi proved that if $Z$ is an irreducible hypersurface in $\mathbb{P}^4(\mathbb{C})$ that contains a three dimensional set of lines, then $Z$ is either a quadratic hypersurface or a scroll of planes. We prove a discretized version of this result for hypersurfaces in $\mathbb{R}^4$. As an application, we prove that at most $δ^{-2-\varepsilon}$ direction-separated $δ$-tubes can be contained in the $δ$-neighborhood of a low-degree hypersurface in $\mathbb{R}^4$. This result leads to improved bounds on the restriction and Kakeya problems in $\mathbb{R}^4$. Combined with previous work of…
The purpose of these notes is describe the state of progress on the restriction problem in harmonic analysis, with an emphasis on the developments of the past decade or so on the Euclidean space version of these problems for spheres and other hypersurfaces. As the field is quite large and has so many applications, it will be impossible to completely survey the field, but we will try to at least give the main ideas and developments in this area. The restriction problem are connected to many other conjectures, notably the Kakeya and Bochner-Riesz conjectures, as well as PDE conjectures such as t…
Consider a trigonometric polynomial f of degree N, and associate to it the polynomial F in which each coefficient of f is replaced by its absolute value. F is called the majorant of f. We show that the L^3 norm of f can be larger than that of F by a power of N. Similar examples could be constructed in which 3 is replaced by any p > 2 not equal to an even integer. This provides a negative answer to a question, the "Hardy-Littlewood majorant problem", which has been the object of some study. Had the answer been affirmative, there would have followed exciting results connected with the restrictio…
We study L^p-L^r restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension $d$ is even, then it is conjectured that the L^{(2d+2)/(d+3)}-L^2 Stein-Tomas restriction result can be improved to the L^{(2d+4)/(d+4)}-L^2 estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured L^p-L^2 restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result…
