Nonempty interior of pinned distance and tree sets

TL;DR

The paper advances the pinned Falconer distance problem by establishing the first nontrivial planar threshold for nonempty interior of pinned distance sets: if $\dim_{\mathcal{H}}(E) > \frac{7}{4}$ in the plane, there exists $y\in E$ with $\Delta^{y}(E)$ containing an interval. It develops an $L^{p}\to L^{p}$ local smoothing framework that converts local smoothing bounds into density and continuity results for pinned distance measures, yielding a general criterion: if $\eta>0$ and a local smoothing estimate holds, then $\dim_{\mathcal{H}}(E)>d-\eta$ ensures interior of the pinned distance set, with two-set extensions and explicit corollaries using recent smoothing bounds. The authors extend these ideas to configurations defined by finite trees, proving that in the plane every finite tree with a pinned vertex yields a pinned interior configuration provided $\dim_{\mathcal{H}}(E)>\frac{7}{4}$; they develop a novel induction on trees using graph maps and density decompositions, leading to new higher-dimensional instances under current local smoothing results. Collectively, the work highlights a robust, $L^{p}\to L^{p}$-based approach to pinned distance and tree problems, linking harmonic analysis, geometric measure theory, and combinatorial graph structures to obtain interior-type results and new dimensional thresholds. Key contributions include (i) a planar nonempty interior threshold for pinned distances, (ii) an explicit local smoothing–driven criterion applicable to distances and to tree-like configurations, and (iii) a systematic induction framework for densities of pinned tree configurations that extends the method to higher dimensions under the best-known local smoothing estimates.

Abstract

For a compact set $E\subset\mathbb{R}^d$, $d\geq 2$, consider the pinned distance set $Δ^{y}(E)=\lbrace |x-y| : x\in E\rbrace$. Peres and Schlag showed that if the Hausdorff dimension of $E$ is bigger than $\frac{d+2}{2}$ with $d\geq 3$, then there exists a point $y\in E$ such that $Δ^{y}(E)$ has nonempty interior. In this paper we obtain the first non-trivial threshold for this problem in the plane, improving on the Peres--Schlag threshold when $d=3$, and we extend the results to trees using a novel induction argument.

Figures (2)

• Figure 1: A sample tree, pinned at $v_0$.
• Figure 2: Decomposition in subtrees for tree in Example \ref{['example tree']}.

Theorems & Definitions (26)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Conjecture 9: Local smoothing conjecture for the wave equation in $\mathbb{R}^{d+1}$ Sogge91
• Theorem 10