A set \(E\) is a set of divergence for the localization problem (SDLP) if there exists \(f\in L^p(\mathbb R^n)\), \(n\geq 2\), with \(f\equiv 0\) in the unit ball, such that the spherical inverse Fourier transform \(S_R f(x)\) diverges for all \(x\in E\) as \(R\rightarrow \infty\), \[ S_R f(x) =\int _{| \xi | \leq R}\widehat f(\xi) e^{2\pi i x\cdot \xi}d\xi, \] where \(\widehat f\) is the Fourier transform of \(f\). It is known that for a set to be SDLP it must have \(n\)-dimensional Lebesgue measure zero, and that singletons are SDLP’s. Some of the new results presented here are:
If a set \(E\subset B\) has Hausdorff dimension less than \(n-1\) , then \(E\) is SDLP.
If \(E\subset E'\times [-1,1]\), where \(E'\) has \(n-1\)-dimensional Lebesgue measure zero, then \(E\) is SDLP.
For a set \(F\subset (0,1)\), let \(E(F)=\{x| | x| \in F\}\). If \(F\subset (0,1)\) is compact and has Hausdorff dimension less than \(1/2\) then \(E(F)\) is SDLP. But if \(F\subset (0,1)\) has Hausdorff dimension greater than \(1/2\), then \(E(F)\) is not an SDLP.
The results presented in this article are proved in the following two articles, A. Carbery and F. Soria [J. Fourier Anal. Appl. 3, Spec. Iss., 847-858 (1997; Zbl 0896.42007)] and A. Carbery, F. Soria and A. Vargas (forthcoming article).