Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Hart, A. Iosevich, D. Koh, and M. Rudnev, “Averages over hyperplanes, sum–product theory in vector spaces over finite fields and the Erdős–Falconer distance conjecture,” Trans. Am. Math. Soc. 363 (6), 3255–3275 (2011).
N. Hegyvári and M. Pálfy, “Note on a result of Shparlinski and related results,” Acta Arith. 193 (2), 157–163 (2020).
D. D. Hieu and L. A. Vinh, “On distance sets and product sets in vector spaces over finite rings,” Mich. Math. J. 62 (4), 779–792 (2013).
A. Iosevich and D. Koh, “Extension theorems for spheres in the finite field setting,” Forum Math. 22 (3), 457–483 (2010).
A. Iosevich, D. Koh, S. Lee, T. Pham, and C.-Y. Shen, “On restriction estimates for the zero radius sphere over finite fields,” Can. J. Math. 73 (3), 769–786 (2021).
A. Iosevich and M. Rudnev, “Erdős distance problem in vector spaces over finite fields,” Trans. Am. Math. Soc. 359 (12), 6127–6142 (2007).
D. Koh, T. Pham, C.-Y. Shen, and L. A. Vinh, “A sharp exponent on sum of distance sets over finite fields,” Math. Z. 297 (3–4), 1749–1765 (2021).
D. Koh, T. Pham, and L. A. Vinh, “Extension theorems and a connection to the Erdős–Falconer distance problem over finite fields,” J. Funct. Anal. 281 (8), 109137 (2021); arXiv: 1809.08699 [math.CA].
R. Lidl and H. Niederreiter, Finite Fields (Cambridge Univ. Press, Cambridge, 1997), Encycl. Math. Appl. 20.
B. Murphy and G. Petridis, “An example related to the Erdős–Falconer question over arbitrary finite fields,” Bull. Hell. Math. Soc. 63, 38–39 (2019).
B. Murphy, G. Petridis, T. Pham, M. Rudnev, and S. Stevens, “On the pinned distances problem over finite fields,” arXiv: 2003.00510 [math.CO].
T. Pham, “Erdős distinct distances problem and extensions over finite spaces,” PhD thesis (EPFL, Lausanne, 2017).
T. Pham and L. A. Vinh, “Distribution of distances in positive characteristic,” Pac. J. Math. 309 (2), 437–451 (2020).
I. E. Shparlinski, “On the additive energy of the distance set in finite fields,” Finite Fields Appl. 42, 187–199 (2016).
Acknowledgments
We are grateful to the reviewers for useful comments and suggestions. We would also like to thank the Vietnam Institute for Advanced Study in Mathematics for hospitality during our visit.
Funding
Daewoong Cheong and Doowon Koh were supported by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A3B07045594 and NRF-2018R1D1A1B07044469, respectively). Thang Pham was supported by the Swiss National Science Foundation grants P400P2-183916 and P4P4P2-191067.
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 314, pp. 290–300 https://doi.org/10.4213/tm4196.
To the 130th anniversary of I. M. Vinogradov
Rights and permissions
About this article
Cite this article
Cheong, D., Koh, D. & Pham, T. An Asymmetric Bound for Sum of Distance Sets. Proc. Steklov Inst. Math. 314, 279–289 (2021). https://doi.org/10.1134/S0081543821040131
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543821040131