arXiv subjects
Publications and source records attributed to José Gaitan.
We find the optimal constant $C$ such that \begin{equation*} \|f_1*f_2*\dots*f_{k}\|_{\infty}\geq C\prod_{i=1}^{k}\|f_i\|_1 \end{equation*} for functions $f_i:\{0,1\}^d\to\mathbb{R}$. As applications, we derive bounds for Sidon sets on hypercubes, and, we also obtain bounds for the continuous analogue problem.
We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, $k$-point configuration sets given by $$Δ^{diag}(E)= \{ \,|(x,x,\dots,x)-(y_1,y_2,\dots,y_{k-1})| : x, y_1, \dots,y_{k-1} \in E\, \}$$ for a compact $E\subset\mathbb{R}^d$ and $k\ge 3$. We show that $Δ^{diag}(E)$ has non-empty interior if the Hausdorff dimension of $E$ satisfies \begin{equation*} \dim(E) > \begin{cases} \frac{2d+1}3, & k=3 \\ \frac{(k-1)d}k,& k\ge 4. \end{cases} \end{equation*} We prove an extension of this to $C^ω$ Riemannian metrics $g$ close to the product of Euclidean metrics. For product metrics this follows from known results on pinned distance sets, but to obtain a result for general perturbations $g$ we present a sequence of proofs of partial results, leading up to the proof of the full result, which is based on estimates for multilinear Fourier integral operators.