论文标题
距离集合总和的不对称结合
An asymmetric bound for sum of distance sets
论文作者
论文摘要
对于$ e \ subset \ mathbb {f} _q^d $，让$δ（e）$表示由$ e $中的点对确定的距离。通过在抛物面，KOH，Pham，Shen和Vinh（2020）上使用套餐的添加能，证明，如果$ e，f \ subset \ mathbb {f} _q^d $是带有$ | e | e | e | e | e | e | e | e | gg q^Q^{d+\ frac {1} $ | e | e | e | e | e | e | e | e | e | e | e | Q/2 $。他们还证明了阈值$ q^{d+\ frac {1} {3}} $在$ | e | = | f | $时很清晰。在本文中，我们在不平衡的情况下提供了这种结果的改进，这在奇数方面基本上是锋利的。我们证明中最重要的工具是零半径范围的最佳$ l^2 $限制定理。
For $ E\subset \mathbb{F}_q^d$, let $Δ(E)$ denote the distance set determined by pairs of points in $E$. By using additive energies of sets on a paraboloid, Koh, Pham, Shen, and Vinh (2020) proved that if $E,F\subset \mathbb{F}_q^d $ are subsets with $|E||F|\gg q^{d+\frac{1}{3}}$ then $|Δ(E)+Δ(F)|> q/2$. They also proved that the threshold $q^{d+\frac{1}{3}}$ is sharp when $|E|=|F|$. In this paper, we provide an improvement of this result in the unbalanced case, which is essentially sharp in odd dimensions. The most important tool in our proofs is an optimal $L^2$ restriction theorem for the sphere of zero radius.
