Geodesic
An Asymmetric Bound for Sum of Distance Sets
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and Combinatorial Number Theory, Tome 314 (2021), pp. 290-300.

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For $E\subset \mathbb F_q^d$, let $\Delta (E)$ 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 $|E|\cdot |F|\gg q^{d+{1}/{3}}$, then $|\Delta (E)+\Delta (F)|>q/2$. They also proved that the threshold $q^{d+{1}/{3}}$ 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. 
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Daewoong Cheong; Doowon Koh; Thang Pham. An Asymmetric Bound for Sum of Distance Sets. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and Combinatorial Number Theory, Tome 314 (2021), pp. 290-300. http://geodesic.mathdoc.fr/item/TM_2021_314_a12/

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