A sharp exponent on sum of distance sets over finite fields

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 F q d , and Δ ( E ) , Δ ( F ) be corresponding distance sets. We prove that if | E | | F | ≥ C q d + 1 3 for a sufficiently large constant C , then the set Δ ( E ) + Δ (...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2021-04, Vol.297 (3-4), p.1749-1765
Main Authors: Koh, Doowon, Pham, Thang, Shen, Chun-Yen, Vinh, Le Anh
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary: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 F q d , and Δ ( E ) , Δ ( F ) be corresponding distance sets. We prove that if | E | | F | ≥ C q d + 1 3 for a sufficiently large constant C , then the set Δ ( E ) + Δ ( 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 F q d , it is shown that the exponent d + 1 3 can be improved to d - 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.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-020-02578-6