论文标题
在奇数维度和应用中绑定的点状发病率
A point-sphere incidence bound in odd dimensions and applications
论文作者
论文摘要
在本文中，我们证明了有限场上的向量空间中绑定的新点晶体发病率。更确切地说，让$ p $是一组点，$ s $是$ \ mathbb {f} _q^d $中的一组球。假设$ | p |，| s | \ le n $，我们证明$ p $和$ s $之间的发病率数量满足\ [i（p，s）\ le n^2q^{ - 1}+q^{\ frac {\ frac {d-1} {2}} {2}}} n，\]在$ d，q $ d，q $和radii下。这改善了已知的上限$ n^2q^{ - 1}+q^{\ frac {d} {2}}} n $。作为一个应用程序，我们表明，对于$ a \ subset \ mathbb {f} _q $，$ q^{1/2} \ ll | a | \ ll q^{\ frac {\ frac {d^2+1} {2d^2}} {2d^2}} $ \ gg \ frac {| a |^d} {q^{\ frac {d-1} {2}}}}。
In this paper, we prove a new point-sphere incidence bound in vector spaces over finite fields. More precisely, let $P$ be a set of points and $S$ be a set of spheres in $\mathbb{F}_q^d$. Suppose that $|P|, |S|\le N$, we prove that the number of incidences between $P$ and $S$ satisfies \[I(P, S)\le N^2q^{-1}+q^{\frac{d-1}{2}}N,\] under some conditions on $d, q$, and radii. This improves the known upper bound $N^2q^{-1}+q^{\frac{d}{2}}N$ in the literature. As an application, we show that for $A\subset \mathbb{F}_q$ with $q^{1/2}\ll |A|\ll q^{\frac{d^2+1}{2d^2}}$, one has \[\max \left\lbrace |A+A|,~ |dA^2|\right\rbrace \gg \frac{|A|^d}{q^{\frac{d-1}{2}}}.\] This improves earlier results on this sum-product type problem over arbitrary finite fields.
