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
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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/

[1] Dao Nguyen Van Anh; Le Quang Ham; Doowon Koh; Thang Pham; Le Anh Vinh On a theorem of Hegyvári and Hennecart, Pac. J. Math., Volume 305 (2020) no. 2, pp. 407-421 | Zbl

[2] Javier Cilleruelo; Alex Iosevich; Ben Lund; Oliver Roche-Newton; Misha Rudnev Elementary methods for incidence problems in finite fields, Acta Arith., Volume 177 (2017) no. 2, pp. 133-142 | DOI | MR | Zbl

[3] Derrick Hart; Alex Iosevich; Doowon Koh; Misha Rudnev Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdős-Falconer distance conjecture, Trans. Am. Math. Soc., Volume 363 (2011) no. 6, pp. 3255-3275 | DOI | Zbl

[4] Alex Iosevich; Doowon Koh Extension theorems for spheres in the finite field setting, Forum Math., Volume 22 (2010) no. 2, pp. 457-483 | MR | Zbl

[5] Alex Iosevich; Doowon Koh; Sujin Lee; Thang Pham; Chun-Yen Shen On restriction estimates for the zero radius sphere over finite fields, Can. J. Math., Volume 73 (2021) no. 3, pp. 769-786 | DOI | MR | Zbl

[6] Alex Iosevich; Misha Rudnev Erdős distance problem in vector spaces over finite fields, Trans. Am. Math. Soc., Volume 359 (2007) no. 12, pp. 6127-6142 | DOI | Zbl

[7] Doowon Koh; Sujin Lee; Thang Pham On the finite field cone restriction conjecture in four dimensions and applications in incidence geometry (2021) (accepted in Int. Math. Res. Not.)

[8] Doowon Koh; Thang Pham; Le Anh Vinh Extension theorems and a connection to the Erdős-Falconer distance problem over finite fields, J. Funct. Anal., Volume 281 (2021) no. 8, 109137, 54 pages | Zbl

[9] Michael Krivelevich; Benny Sudakov Pseudo-random graphs, More sets, graphs and numbers (Bolyai Society Mathematical Studies), Volume 15, Springer, 2006, pp. 199-262 | DOI | MR

[10] Rudolf Lidl; Harald Niederreiter Finite fields, Encyclopedia of Mathematics and Its Applications, 20, Cambridge University Press, 1996 | DOI

[11] Ali Mohammadi; Sophie Stevens Attaining the exponent $5/4$ for the sum-product problem in finite fields (2021) (https://arxiv.org/abs/2103.08252)

[12] Duc Hiep Pham A note on sum-product estimates over finite valuation rings, Acta Arith., Volume 198 (2021) no. 2, pp. 187-194 | DOI | MR | Zbl

[13] Nguyen D. Phuong; Pham Thang; Le Anh Vinh Incidences between points and generalized spheres over finite fields and related problems, Forum Math., Volume 29 (2017) no. 2, pp. 449-456 | DOI | MR | Zbl

[14] Misha Rudnev; Ilya D. Shkredov; Sophie Stevens On the energy variant of the sum-product conjecture, Rev. Mat. Iberoam., Volume 36 (2019) no. 1, pp. 207-232 | DOI | MR | Zbl

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