A point-sphere incidence bound in odd dimensions and applications

Koh, Doowon; Pham, Thang

Comptes Rendus. Mathématique, Tome 360 (2022) no. G6, 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 $\left|P\right|,\left|S\right|\le N$, we prove that the number of incidences between $P$ and $S$ satisfies

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

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 $q^{1/2}\ll \left|A\right|\ll q^{\frac{d^2+1}{2d^2}}$, one has

$$max\left\{|A+A|,|d{A}^2|\right\}\gg \frac{{\left|A\right|}^d}{q^{\frac{d-1}{2}}}.$$

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

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

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

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

[5] Iosevich, Alex; Koh, Doowon; Lee, Sujin; Pham, Thang; Shen, Chun-Yen 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] Iosevich, Alex; Rudnev, Misha Erdős distance problem in vector spaces over finite fields, Trans. Am. Math. Soc., Volume 359 (2007) no. 12, pp. 6127-6142 | DOI | Zbl

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

[8] Koh, Doowon; Pham, Thang; Vinh, Le Anh 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] Krivelevich, Michael; Sudakov, Benny Pseudo-random graphs, More sets, graphs and numbers (Bolyai Society Mathematical Studies), Volume 15, Springer, 2006, pp. 199-262 | DOI | MR

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

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

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

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