Related papers: Large sets without Fourier restriction theorems

For any $0 < \alpha <1$, we construct Cantor sets on the parabola of Hausdorff dimension $\alpha$ such that they are Salem sets and each associated measure $\nu$ satisfies the estimate $\|{\widehat{f d\nu}}\|_{L^p(\mathbb{R}^2)} \leq C_p…

For any $\alpha\in(0,d)$, we construct Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $\alpha$ such that the associated natural measure $\mu$ obeys the restriction estimate $\| \widehat{f d\mu} \|_{p} \leq C_p \| f \|_{L^2(\mu)}$ for…

For $2\leq p<\infty$, $\alpha'>2/p$, and $\delta>0$, we construct Cantor-type measures on $\mathbb{R}$ supported on sets of Hausdorff dimension $\alpha<\alpha'$ for which the associated maximal operator is bounded from $L^p_\delta…

A famous theorem of Carleson says that, given any function $f\in L^p(\TT)$, $p\in(1,+\infty)$, its Fourier series $(S_nf(x))$ converges for almost every $x\in \mathbb T$. Beside this property, the series may diverge at some point, without…

For $0 < \alpha \leq 1$, let $E$ be a compact subset of the $d$-dimensional moment curve in $\mathbb{R}^d$ such that $N(E,\varepsilon) \lesssim \varepsilon^{-\alpha}$ for $0 <\varepsilon <1$ where $N(E,\varepsilon)$ is the smallest number…

We study maximal averages associated with singular measures on $\rr$. Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension $1 - \epsilon$, $0 \leq \epsilon < {1/3}$ for which the…

For $\lambda\in(0,1/2]$ let $K_\lambda \subset\mathbb{R}$ be a self-similar set generated by the iterated function system $\{\lambda x, \lambda x+1-\lambda\}$. Given $x\in(0,1/2)$, let $\Lambda(x)$ be the set of $\lambda\in(0,1/2]$ such…

For $\lambda\in(0,1/3]$ let $C_\lambda$ be the middle-$(1-2\lambda)$ Cantor set in $\mathbb R$. Given $t\in[-1,1]$, excluding the trivial case we show that \[ \Lambda(t):=\left\{\lambda\in(0,1/3]:…

We prove $L^p$ bounds for the maximal operators associated to an Ahlfors-regular variant of fractal percolation. Our bounds improve upon those obtained by I. {\L}aba and M. Pramanik and in some cases are sharp up to the endpoint. A…

Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\Bbb R^n$. If $X$ contains a ball of each…

We use a deterministic construction to prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem for dimension $d=1$ and parameter range $0 < a,b \leq d$ and $b\leq 2a$. Previous constructions by…

In this article we prove the existence of sets $E \subseteq \mathbb{R}$ of zero Fourier dimension such that it is possible to restrict the Fourier transform to $E$ on a certain non-trivial range $[1,\tilde{p})$ with $1<\tilde{p}<2$. This…

For $0<\alpha<1$ let $V(\alpha)$ denote the supremum of the numbers $v$ such that every $\alpha$-H\"older continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2…

Fourier restriction theorems, whose study had been initiated by E.M. Stein, usually describe a family of a priori estimates of the L^q-norm of the restriction of the Fourier transform of a function f in L^p (say, on Euclidean space) to a…

We prove a maximal Fourier restriction theorem for the sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^{d}$ for any dimension $d\geq 3$ in a restricted range of exponents given by the Stein-Tomas theorem. The proof consists of a simple…

Let ${{\bf R}_{\mathbb{S}^{d-1}}}(p\to q)$ denote the best constant for the $L^p(\mathbb{R}^d)\to L^q(\mathbb{S}^{d-1})$ Fourier restriction inequality to the unit sphere $\mathbb{S}^{d-1}$, and let ${\bf R}_{\mathbb{S}^{d-1}} (p\to…

A conjecture of Erd\H{o}s states that for any infinite set $A \subseteq \mathbb R$, there exists $E \subseteq \mathbb R$ of positive Lebesgue measure that does not contain any nontrivial affine copy of $A$. The conjecture remains open for…

We investigate variants of the Erd\H{o}s similarity problem for Cantor sets. We prove that under a mild Hausdorff or packing logarithmic dimension assumption, Cantor sets are not full measure universal, significantly improving the known…

Let $m\in\mathbb N_{\ge 2}$, and let $\mathcal K=\{K_\lambda: \lambda\in(0, 1/m]\}$ be a class of Cantor sets, where $K_{\lambda}=\{\sum_{i=1}^\infty d_i\lambda^i: d_i\in\{0,1,\ldots, m-1\}, i\ge 1\}$. We investigate in this paper the…

We study the Falconer distance set problem in Euclidean space and obtain improved dimensional estimates under natural Fourier analytic assumptions cast in terms of the Fourier dimension and spectrum. Interestingly, under reasonably mild…