arXiv subjects
Publications and source records attributed to Donggeun Ryou.
We consider multifractal Mandelbrot cascades supported on planar $C^2$ curves with nonvanishing curvature and show that their Fourier dimension is as large as possible, i.e., equal to the infimum of the lower pointwise dimension of the measure.
For $0 < α\leq 1$, let $E$ be a compact subset of the $d$-dimensional moment curve in $\mathbb{R}^d$ such that $N(E,\varepsilon) \lesssim \varepsilon^{-α}$ for $0 <\varepsilon <1$ where $N(E,\varepsilon)$ is the smallest number of $\varepsilon$-balls needed to cover $E$. We proved that if $f \in L^p(\mathbb{R}^d)$ with \begin{align*} 1 \leq p\leq p_α:= \begin{cases} \frac{d^2+d+2α}{2α} & d \geq 3, \frac{4}α &d =2, \end{cases} \end{align*} and $\widehat{f}$ is supported on the set $E$, then $f$ is identically zero. We also proved that the range of $p$ is optimal by considering random Cantor sets on the moment curve. We extended the result of Guo, Iosevich, Zhang and Zorin-Kranich, including the endpoint. We also considered applications of our results to the failure of the restriction estimates and Wiener Tauberian Theorem.
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 Hambrook and Łaba \cite{HL2013} and Chen \cite{chen} required randomness and only covered the range $0 < b \leq a \leq d=1$. We also resolve a question of Seeger \cite{seeger-private} about the Fourier restriction inequality on the sets of well-approximable numbers.
We prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions $d$ and the full parameter range $0 < a,b < d$. Our construction is deterministic and also yields Salem sets.
We construct maximal $Λ(p)$-subsets on a large class of curved manifolds, in an optimal range of Lebesgue exponents $p$. Our arguments combine restriction estimates and decoupling with old and new probabilistic estimates.
This article is a study guide for ``On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane" by Orponen and Shmerkin. We begin by introducing Furstenberg set problem and exceptional set of projections and provide a summary of the proof with the core ideas.
For any $0 < α<1$, we construct Cantor sets on the parabola of Hausdorff dimension $α$ such that they are Salem sets and each associated measure $ν$ satisfies the estimate $\|{\widehat{f dν}}\|_{L^p(\mathbb{R}^2)} \leq C_p \|{f}\|_{L^2(ν)}$ for all $p >6/α$ and for some constant $C_p >0$ which may depend on $p$ and $ν$. The range $p>6/α$ is optimal except for the endpoint. The proof is based on the work of Laba and Wang on restriction estimates for random Cantor sets and the work of Shmerkin and Suomala on Fourier decay of measures on random Cantor sets. They considered fractal subsets of $\mathbb{R}^d$, while we consider fractal subsets of the parabola.
We introduce $ Λ(Φ) $-sets as generalizations of $ Λ(p) $-sets. These sets are defined in terms of Orlicz norms. We consider $Λ(Φ)$-sets when the Matuszewska-Orlicz index of $ Φ$ is larger than $ 2 $. When $S$ is a $Λ(Φ)$-set, we establish an estimate of the size of $ S \cap [-N,N] $ where $ N \in \mathbb{N} $. Next, we construct a $ Λ(Φ_1)$-set which is not a $ Λ(Φ_2)$-set for any $ Φ_2 $ such that $ \sup_{u \geq 1} Φ_2(u) / Φ_1(u) = \infty $ by using a probabilistic method. With an additional assumption about a subset $E$ of $\mathbb{Z}$, we can construct such a $Λ(Φ_1)$-set contained in $E$. These statements extend known results on the structure of $ Λ(p) $-sets to $Λ(Φ)$-sets.