arXiv subjects

Sam Craig

Publications and source records attributed to Sam Craig.

2 recordsLinked to original sources

arXiv2025

The quasi-Assouad dimension of $(1,2t)$-Furstenberg sets in $\mathbb{R}^3$ is extremized by sticky sets

A $(1,2t)$-Furstenberg set in $\mathbb{R}^3$ is naturally defined as a set containing a union of unit line segments forming a $2t$-dimensional subset of the affine Grassmannian in $\mathbb{R}^3$ and satisfying a suitable variant of the Frostman Convex Wolff Axiom. Some of these sets have a multi-scale self-similarity property called stickiness. We investigate the extremizers of the quasi-Assouad dimension of $(1,2t)$-Furstenberg sets, a slightly stronger variant of the Assouad dimension. We prove that sticky $(1,2t)$-Furstenberg sets have the least possible quasi-Assouad dimension among all $(1,2t)$-Furstenberg sets. This result also follows from Corollary 1.10 of Wang and Zahl's solution to the Kakeya conjecture, which implies that all $(1,2t)$-Furstenberg sets have Hausdorff dimension $2t+1$.

math.CA↗

arXiv2024

Failure of weak-type endpoint restriction estimates for quadratic manifolds

It is well-known that the Fourier extension operator for the paraboloid in $\mathbb{R}^d$ cannot be weak-type bounded at the restriction endpoint $q = 2d/(d-1)$, since such an estimate would imply bounds for the Kakeya maximal function which contradict the existence of Besicovitch sets. We generalize this approach to prove that the Fourier extension operator for an $n$-dimensional quadratic manifold $\mathcal{M}$ cannot be weak-type bounded at the restriction endpoint. The key step in this proof is constructing a set $K \subset \mathbb{R}^d$ containing a translate of every plane normal to $\mathcal{M}$ which can be covered by $\lesssim \delta^{-d}\left(\frac{\log \log (1/\delta)}{\log (1/\delta)}\right)^{n/(d-n)}$ many $\delta$-balls. Such a set rules out endpoint bounds for the associated Kakeya maximal function and hence weak-type endpoint estimates for the restriction operator.

math.CA↗