arXiv subjects
Publications and source records attributed to Terence L. J. Harris.
It is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension $\dim A>1$, and if $ρ_θ$ is orthogonal projection to the line spanned by $( \cos θ, \sin θ, 1 )$, then $ρ_θ(A)$ has positive length for all $θ$ outside a set of Hausdorff dimension at most $\frac{3-\dim A}{2}$.
It is shown that if $u$ solves the wave equation in $\mathbb{R}^{4+1}$ with initial data $u(\cdot,0) = u_0(\cdot) \in H^s$ and $\partial_tu(\cdot,0) = u_1(\cdot ) \in H^{s-1}$, where $0.5 < s \leq 0.55$, then $u(x,t) \to u_0(x)$ and $\partial_tu(x,t) \to u_1(x)$ pointwise as $t \to 0$, for all $x$ outside an exceptional set of Hausdorff dimension at most $6-4s$. In a very small range of $s$, this verifies a conjecture of Barceló, Bennett, Carbery, and Rogers. More generally, a partial improvement to the exceptional set bound in $\mathbb{R}^{n+1}$ is obtained for $n \geq 4$ and $1/2 < s < n/4$.
For Borel sets $A\subset \mathbb{R}^n\times \mathbb{R}$ we prove lower bounds for the parabolic Hausdorff dimension of the orthogonal projections of $A$ on generic $m$-dimensional linear subspaces of $\mathbb{R}^n\times \mathbb{R}$.
It is shown that Schrödinger maximal inequalities over fractals are equivalent to the $L^2$ decay rates of Fourier transforms of fractal measures over the paraboloid. A similar connection is shown between the wave equation and cone averages. One implication is well-known and follows from the Kolmogorov-Seliverstov-Plessner method, but the other implication is nontrivial and relies on a variant of the Marstrand projection theorem. The idea of the proof is to insert an extra averaging parameter into a proof of Lucà and Rogers, which used a quantitative ergodic lemma instead of the Marstrand projection theorem. Lucà and Rogers gave a second proof of Bourgain's necessary condition $s\geq \frac{n}{2(n+1)}$ for Schrödinger solutions in $\mathbb{R}^{n+1}$ to converge pointwise a.e. back to the initial data as time tends to zero. One application of the main theorem in this article is a proof of Bourgain's necessary condition which does not use ergodic theory or number theory.
It is shown that if $A$ is a Borel subset of the first Heisenberg group, with Hausdorff dimension satisfying $2< \dim A < 3$, then the packing dimensions of vertical projections of $A$ are almost surely not less than $\dim A$, where both packing and Hausdorff dimensions are defined with respect to the Korányi metric. For the Hausdorff dimension of the projections, a weaker almost sure lower bound is obtained which improves the known bound in the range $2 < \dim A < \frac{1}{8}\left( 17 + \sqrt{33}\right) \approx 2.84$. The bound is slightly larger than $1+\frac{1}{2} \dim A$ and behaves similarly near $\dim A =2$. Both proofs rely on a variable coefficient local smoothing inequality.
We prove bounds on approximate incidences between families of circles and families of points in the plane. As a consequence, we prove a lower bound for the dimension of circular $(u,v)$-Furstenberg sets, which is new for large $u$ and $v$.
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 $V_θ=γ(θ)^\perp$ and let $π_θ:\mathbb{R}^3\rightarrow V_θ$ be the orthogonal projections. We prove that if $A\subset \mathbb{R}^3$ is a Borel set, then for a.e. $θ\in [0,1]$ we have $\text{dim}(π_θ(A))=\min\{2,\text{dim} A\}$. More generally, we prove an exceptional set estimate. For $A\subset\mathbb{R}^3$ and $0\le s\le 2$, define $E_s(A):=\{θ\in[0,1]: \text{dim}(π_θ(A)) 2$, then for a.e. $θ\in[0,1]$ we have $\mathcal{H}^2(π_θ(A))>0$.
It is shown that $SL_2$ Besicovitch sets of measure zero exist in $\mathbb{R}^3$. The proof is constructive and uses point-line duality analogously to Kahane's construction of measure zero Besicovitch sets in the plane. A corollary is that the $SL_2$ Kakeya maximal inequality cannot hold with uniform constant. A counterexample is given to show that the $SL_2$ Kakeya maximal inequality cannot hold for $p> 3/2$; even in the model case where the $δ$-tubes have $δ$-separated directions and the cardinality of the tube family is $\sim δ^{-2}$. It is then shown that, with $C_ε δ^{-ε}$ loss, the $SL_2$ Kakeya maximal inequality does hold if $p \leq 4/3$, whenever the tubes satisfy a 2-dimensional ball condition (equivalent to the Wolff axioms in the $SL_2$ case). The proof is via an $L^{4/3}$ inequality for restricted families of projections onto planes. For both inequalities, the range $4/3 < p \leq 3/2$ remains an open problem. A related $L^{6/5}$ inequality is derived for restricted projections onto lines. Finally, an application is given to generic intersections of sets in $\mathbb{R}^3$ with "light rays" and "light planes".
It is shown that vertical projections in the Heisenberg group of sets of dimension strictly greater than 3 almost surely have positive area. The proof uses the point-plate incidence method introduced by Fässler and Orponen, and also uses a similar approach to a recent maximal inequality of Zahl for fractal families of tubes. It relies on the endpoint trilinear Kakeya inequality in $\mathbb{R}^3$. Some related results are given on generic intersections with horizontal lines.
Let $γ: I \to S^2$ be a $C^2$ curve with $\det(γ, γ', γ'')$ nonvanishing, and for each $θ\in I$ let $ρ_θ$ be orthogonal projection onto the span of $γ(θ)$. It is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension strictly greater than 1, then $ρ_θ(A)$ has positive length for a.e. $θ\in I$. This answers a question raised by Käenmäki, Orponen and Venieri.
An improved a.e. lower bound is given for Hausdorff dimension under vertical projections in the first Heisenberg group, with respect to the Carnot-Carathéodory metric. This improves the known lower bound, and answers a question of Fässler and Hovila. The approach uses the Euclidean Fourier transform, Basset's integral formula, and modified Bessel functions of the second kind.
It is shown that if $γ: [a,b] \to S^2$ is $C^3$ with $\det(γ, γ', γ'') \neq 0$, and if $A \subseteq \mathbb{R}^3$ is a Borel set, then $\dim π_θ (A) \geq \min\left\{ 2,\dim A, \frac{ \dim A}{2} + \frac{3}{4} \right\}$ for a.e. $θ\in [a,b]$, where $π_θ$ denotes projection onto the orthogonal complement of $γ(θ)$ and ``$\dim$'' refers to Hausdorff dimension. This partially resolves a conjecture of Fässler and Orponen in the range $1< \dim A \leq 3/2$, which was previously known only for non-great circles. For $3/2 < \dim A < 5/2$ this improves the known lower bound for this problem.
It is shown that the cone in $\mathbb{R}^{d+1}$ has Fourier dimension $d-1$. This verifies a conjecture of Fraser and Kroon.
A $(d,k)$-set is a subset of $\mathbb{R}^d$ containing a $k$-dimensional unit ball of all possible orientations. Using an approach of D.~Oberlin we prove various Fourier dimension estimates for compact $(d,k)$-sets. Our main interest is in restricted $(d,k)$-sets, where the set only contains unit balls with a restricted set of possible orientations $Γ$. In this setting our estimates depend on the Hausdorff dimension of $Γ$ and can sometimes be improved if additional geometric properties of $Γ$ are assumed. We are led to consider cones and prove that the cone in $\mathbb{R}^{d+1}$ has Fourier dimension $d-1$, which may be of interest in its own right.
An inequality is derived for the average $t$-energy of pinned distance measures, where $0 < t < 1$. This refines Mattila's theorem on distance sets to pinned distance sets, and gives an analogue of Liu's theorem for pinned distance sets of dimension smaller than 1.
It is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension $\dim A \in (3/2,5/2)$, then for a.e. $θ\in [0,2π)$ the projection $π_θ(A)$ of $A$ onto the 2-dimensional plane orthogonal to $\frac{1}{\sqrt{2}}(\cos θ, \sin θ, 1)$ satisfies $\dim π_θ(A) \geq \max\left\{\frac{4\dim A}{9} + \frac{5}{6},\frac{2\dim A+1}{3} \right\}$. This improves the bound of Oberlin and Oberlin, and of Orponen and Venieri, for $\dim A \in (3/2,5/2)$. More generally, a weaker lower bound is given for families of planes in $\mathbb{R}^3$ parametrised by curves in $S^2$ with nonvanishing geodesic curvature.
We study the family of vertical projections whose fibers are right cosets of horizontal planes in the Heisenberg group, $\mathbb{H}^n$. We prove lower bounds for Hausdorff dimension distortion of sets under these mappings, with respect to the Euclidean metric and also the natural quotient metric which we show behaves like the Euclidean metric in this context. Our bounds are sharp in a large part of the dimension range, and we give conjectural sharp lower bounds for the remaining range. Our approach also lets us improve the known almost sure lower bound for the standard family of vertical projections in $\mathbb{H}^n$ for $n \geq 2$.
Some analogues of the Schrödinger refined Strichartz inequalities (Du, Guth, Li and Zhang) are obtained for the wave equation. These are used to improve the best known $L^2$ fractal Strichartz inequalities for the wave equation in dimensions $d \geq 4$.