For a compact subset \(E\subset\mathbb{R}^d\), define its distance set by \[ \Delta(E) := \{|x-y| : x,y\in E\}. \] For \(d\geq 3\), the authors prove that if \[ \dim E > \alpha, \quad\text{where}\quad\alpha:= \begin{cases} 1.8, & d=3\\ \frac{d}{2} +\frac14+\frac{d+1}{4(2d+1)(d-1)}, & d \geq 4, \end{cases} \] then \(|\Delta (E)| > 0\), where \(\dim E\) denotes the Hausdorff dimension of \(E\) and \(|\cdot|\) Lebesgue measure.
The proof relies on some weighted Fourier restriction estimates and employs polynomial partitioning and refined Strickartz estimates.