Combinatorics
A point-sphere incidence bound in odd dimensions and applications
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 687-698.

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 ${𝔽}_{q}^{d}$. Suppose that $|P|,|S|\le N$, we prove that the number of incidences between $P$ and $S$ satisfies

 $I\left(P,S\right)\le {N}^{2}{q}^{-1}+{q}^{\frac{d-1}{2}}N,$

under some conditions on $d,q$, and radii. This improves the known upper bound ${N}^{2}{q}^{-1}+{q}^{\frac{d}{2}}N$ in the literature. As an application, we show that for $A\subset {𝔽}_{q}$ with ${q}^{1/2}\ll |A|\ll {q}^{\frac{{d}^{2}+1}{2{d}^{2}}}$, one has

 $max\left\{|A+A|,\phantom{\rule{3.33333pt}{0ex}}|d{A}^{2}|\right\}\gg \frac{{|A|}^{d}}{{q}^{\frac{d-1}{2}}}.$

This improves earlier results on this sum-product type problem over arbitrary finite fields.

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.333
Doowon Koh 1; Thang Pham 2

1 Department of Mathematics, Chungbuk National University, Korea
2 University of Science, Vietnam National University, Hanoi, Vietnam
License: CC-BY 4.0
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Doowon Koh; Thang Pham. A point-sphere incidence bound in odd dimensions and applications. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 687-698. doi : 10.5802/crmath.333. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.333/

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