Related papers: Kakeya sets of curves

We prove a bilinear Kakeya inequality in the first Heisenberg group and a sharp bilinear Kakeya estimate for Euclidean curved tubes in $\R^2$. By adapting an argument of F\"assler, Pinamonti and Wald involving Heisenberg projections, we…

We prove that in all dimensions at least 3 and for any H\"ormander-type oscillatory integral operator satisfying Bourgain's condition, the sticky case of the corresponding curved Kakeya conjecture reduces to the sticky case of the classical…

We prove that every Kakeya set in $\mathbb{R}^3$ formed from lines of the form $(a,b,0) + \operatorname{span}(c,d,1)$ with $ad-bc=1$ must have Hausdorff dimension $3$; Kakeya sets of this type are called $SL_2$ Kakeya sets. This result was…

We give new lower bounds for the Hausdorff dimension of Kakeya sets built from various families of curves in $\mathbb R^3$, going beyond what the polynomial partitioning method has so-far achieved. We do this by combining Wolff's classical…

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…

Around the early 2000-s, Bourgain, Katz and Tao introduced an arithmetic approach to study Kakeya-type problems. They showed that the Euclidean Kakeya conjecture follows from a natural problem in additive combinatorics, now referred to as…

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…

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…

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…

The purpose of this paper is to investigate the well-posedness of several linear and nonlinear equations with a parabolic forward-backward structure, and to highlight the similarities and differences between them. The epitomal linear…

We study a variety of problems about homothets of sets related to the Kakeya conjecture. In particular, we show many of these problems are equivalent to the arithmetic Kakeya conjecture of Katz and Tao. We also provide a proof that the…

We show that for each odd integer $n\ge 3$, there is an open dense subset of H\"ormander phase functions in $\mathbb{R}^n$ for which the associated curved Kakeya sets have Hausdorff dimension at least $\frac{n+1}{2} + d_n$ for some positive…

This paper presents several new results related to the Kakeya problem. First, we establish a geometric inequality which says that collections of direction-separated tubes (thin neighborhoods of line segments that point in different…

Roughly speaking, the Kakeya Conjecture asks to what extent lines which point in different directions can be packed together in a small space. In $\R^2$, the problem is relatively straightforward and was settled in the 1970s. In $\R^3$ it…

We develop an algebraic version of Cartan method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space $W$ with respect to the action of a subgroup $G$ of the $GL(W)$. Under…

In this paper, we study curved Kakeya sets associated with phase functions satisfying Bourgain's condition. In particular, we show that the analysis of curved Kakeya sets arising from translation-invariant phase functions under Bourgain's…

We consider the Cauchy problem and the source problem for normally hyperbolic operators on the Minkowski spacetime, and study the determination of solutions from their integrals along null geodesics. For the Cauchy problem, we give a new…

In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every direction. A famous conjecture, known as…

In this paper, we study the Kuralay equations, namely, the Kuralay-I equation (K-IE) and the Kuralay-II equation (K-IIE). The integrable motion of space curves induced by these equations is investigated. The gauge equivalence between these…

This paper shows that the well-known curve optimization problems which lead to the straight line, the catenary curve, the brachistochrone, and the circle, can all be handled using a unified formalism. Furthermore, from the general…