Two measurable sets $S,\unicode[STIX]{x1D6EC}\subseteq \mathbb{R}^{d}$ form a Heisenberg uniqueness pair, if every bounded measure $\unicode[STIX]{x1D707}$ with support in $S$ whose Fourier transform vanishes on $\unicode[STIX]{x1D6EC}$ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathbb{R}^{d}$. As a corollary we obtain a new, surprising version of the classical Cramér–Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of a dense set of projections). We also give an application to the unique continuation of eigenfunctions of second-order PDEs with constant coefficients.

中文翻译：

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关于二次曲面和海森堡唯一性对的 CRAMÉR-WOLD 定理

两个可测量的集合$S,\unicode[STIX]{x1D6EC}\subseteq \mathbb{R}^{d}$形成一个海森堡唯一性对，如果每个有界测度$\unicode[STIX]{x1D707}$在支持下$新元其傅里叶变换消失$\unicode[STIX]{x1D6EC}$必须为零。我们证明了一个二次超曲面和两个超平面在一般位置上的并集形成了一个海森堡唯一性对$\mathbb{R}^{d}$. 作为推论，我们得到了一个新的、令人惊讶的经典Cramér-Wold定理：二次超曲面上支持的有界测度由其在两个通用超平面上的投影唯一确定（而任意测度需要了解一组稠密的投影） ）。我们还应用了具有常数系数的二阶 PDE 的特征函数的唯一续延。