Abstract
We study a variant of the Erdős–Falconer distance problem in the setting of finite fields. More precisely, let E and F be sets in \(\mathbb {F}_q^d\), and \(\Delta (E), \Delta (F)\) be corresponding distance sets. We prove that if \(|E||F|\ge Cq^{d+\frac{1}{3}}\) for a sufficiently large constant C, then the set \(\Delta (E)+\Delta (F)\) covers at least a half of all distances. Our result in odd dimensional spaces is sharp up to a constant factor. When E lies on a sphere in \({\mathbb {F}}_q^d,\) it is shown that the exponent \(d+\frac{1}{3}\) can be improved to \(d-\frac{1}{6}.\) Finally, we prove a weak version of the Erdős–Falconer distance conjecture in four-dimensional vector spaces for multiplicative subgroups over prime fields. The novelty in our method is a connection with additive energy bounds of sets on spheres or paraboloids.
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References
Chapman, J., Erdoğan, M.Burak, Hart, D., Iosevich, A., Koh, D.: Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates. Math. Z. 271(1), 63–93 (2012)
Hart, D., Iosevich, A., Koh, D., Rudnev, M.: 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)
Hegyvári, N., Pálfy, M.: Note on a result of Shparlinski and related results. Acta Arith. (2019)
Hegyvári, N., Hennecart, F.: A note on Freiman models in Heisenberg groups. Isr. J. Math. 189(1), 397–411 (2012)
Iosevich, A., Koh, D., Lewko, M.: Finite field restriction estimates for the paraboloid in high even dimensions. J. Funct. Anal. (2019)
Iosevich, A., Rudnev, M.: Erdős distance problem in vector spaces over finite fields. Trans. Am. Math. Soc. 359, 6127–6142 (2007)
Iosevich, A., Shparlinski, I., Xiong, M.: Sets with integral distances in finite fields. Trans. Am. Math. Soc. 362(4), 2189–2204 (2010)
Iosevich, A., Koh, D., Lee, S., Pham, T., Shen, C.-Y.: On restriction estimates for the zero radius sphere over finite fields. Can. J. Math. arXiv:1806.11387 (2018)
Hieu, D.D., Vinh, L.A.: On distance sets and product sets in vector spaces over finite rings. Mich. Math. J. 62, 779–792 (2013)
Koh, D., Pham, T., Vinh, L.A.: Extension theorems and a connection to the Erdős–Falconer distance problem over finite fields. arXiv:1809.08699 (preprint) (2018)
Macourt, S., Shkredov, I. D., Shparlinski, I.: Multiplicative energy of shifted sugroups and bounds on exponential sums with trinomials in finite fields. Can. J. Math. (2018)
Murphy, B., Petridis, G., Roche-Newton, O., Rudnev, M., Shkredov, I.D.: New results on sum-product type growth over fields. Mathematika 65(3), 588–642 (2019)
Murphy, B., Petridis, G.: An example related to the Erdős–Falconer question over arbitrary finite fields. Bull. Hellenic Math. Soc. 63, 38–39 (2019). (preprint)
Murphy, B., Petridis, G., Pham, T., Rudnev, M., Stevens, S.: On the pinned distances problem over finite fields. arXiv:2003.00510 (2020)
Pham, T.: Erdős distinct distances problem and extensions over finite spaces. Phd thesis, EPFL, Lausane, Switzerland (2017)
Pham, T., Vinh, L.A., De Zeeuw, F.: Three-variable expanding polynomials and higher-dimensional distinct distances. Combinatorica 39(2), 411–426 (2019)
Rudnev, M., Shkredov, I.D.: On the restriction problem for discrete paraboloid in lower dimension. Adv. Math. 339, 657–671 (2018)
Shparlinski, I.E.: On the additive energy of the distance set in finite fields. Finite Fields Appl. 42, 187–199 (2016)
Acknowledgements
The authors would like to thank Igor Shparlinski for useful comments and suggestions. The authors thank Vietnam Institute for Advanced Study in Mathematics for the hospitality during their visit in Feb 2020.
The authors also would like to thank the referee for useful comments and suggestions.
Doowon Koh was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MIST) (No. NRF-2018R1D1A1B07044469). Thang Pham was supported by Swiss National Science Foundation Grant P400P2-183916. Chun-Yen Shen was supported in part by MOST, through Grant 108-2628-M-002-010-MY4. Le Anh Vinh was supported by the National Foundation for Science and Technology Development Project 101.99-2019.318.
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Koh, D., Pham, T., Shen, CY. et al. A sharp exponent on sum of distance sets over finite fields. Math. Z. 297, 1749–1765 (2021). https://doi.org/10.1007/s00209-020-02578-6
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DOI: https://doi.org/10.1007/s00209-020-02578-6