## 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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## 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, C. *et al.* A sharp exponent on sum of distance sets over finite fields.
*Math. Z.* (2020). https://doi.org/10.1007/s00209-020-02578-6

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