The first purpose of this paper is to provide new finite field extension theorems for paraboloids and spheres. By using the unusual good Fourier transform of the zero sphere in some specific dimensions, which has been discovered recently in the work of Iosevich, Lee, Shen, and the first and second listed authors (2018), we provide new $L^2\to L^r$ extension estimates for paraboloids in certain odd dimensions with −1 non-square, which improves significantly the recent exponent obtained by the first listed author. In the case of spheres, we introduce a way of using the first association scheme graph to analyze energy sets, and as a consequence, we obtain new $L^p\to L^4$ extension theorems for spheres of primitive radii in odd dimensions, which break the Stein-Tomas result toward $L^p\to L^4$ which has stood for more than ten years. Most significantly, it follows from the results for spheres that there exists a different extension phenomenon between spheres and paraboloids in odd dimensions, namely, the $L^p\to L^4$ estimates for spheres with primitive radii are much stronger than those for paraboloids. The second purpose is to show that there is a connection between the restriction conjecture associated to paraboloids and the Erdős-Falconer distance conjecture over finite fields. The last is to prove that the Erdős-Falconer distance conjecture holds in odd dimensional spaces when we study distances between two sets: one set lies on a variety (a paraboloid or a sphere), and the other set is arbitrary in vector spaces over finite fields.

本文的第一个目的是为抛物面和球体提供新的有限域扩展定理。通过使用最近在 Iosevich、Lee、Shen 以及第一和第二作者 (2018) 的工作中发现的某些特定维度中零球面的异常好傅立叶变换，我们提供了新的${升}^2\to {升}^r$某些奇数维度中抛物面的扩展估计具有 -1 非平方，这显着改善了第一作者获得的最近指数。在球体的情况下，我们引入了一种使用第一个关联方案图来分析能量集的方法，因此，我们获得了新的${升}^{磷}\to {升}^4$ 奇维原始半径球体的扩展定理，它打破了 Stein-Tomas 结果 ${升}^{磷}\to {升}^4$已经站了十多年了。最重要的是，从球体的结果可以看出，奇维数的球体和抛物面之间存在不同的扩展现象，即${升}^{磷}\to {升}^4$对具有原始半径的球体的估计比对抛物面的估计强得多。第二个目的是表明与抛物面相关的限制猜想与有限域上的 Erdős-Falconer 距离猜想之间存在联系。最后是证明 Erdős-Falconer 距离猜想在我们研究两个集合之间的距离时在奇维空间中成立：一个集合位于各种（抛物面或球体）上，另一个集合在有限的向量空间中是任意的领域。