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 $$\mathbb {F}_q^d$$ F q d , and $$\Delta (E), \Delta (F)$$ Δ ( E ) , Δ ( F ) be corresponding distance sets. We prove that if $$|E||F|\ge Cq^{d+\frac{1}{3}}$$ | E | | F | ≥ C q d + 1 3 for a sufficiently large constant C , then the set $$\Delta (E)+\Delta (F)$$ 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 $${\mathbb {F}}_q^d,$$ F q d , it is shown that the exponent $$d+\frac{1}{3}$$ d + 1 3 can be improved to $$d-\frac{1}{6}.$$ 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.

中文翻译：

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有限域上距离集总和的尖锐指数

我们研究了有限域设置中 Erdős-Falconer 距离问题的一个变体。更准确地说，让 E 和 F 设置在 $$\mathbb {F}_q^d$$ F qd ，和 $$\Delta (E), \Delta (F)$$ Δ ( E ) , Δ ( F )是对应的距离集。我们证明如果 $$|E||F|\ge Cq^{d+\frac{1}{3}}$$ | E | | F | ≥ C qd + 1 3 对于足够大的常数 C ，则集合 $$\Delta (E)+\Delta (F)$$ Δ ( E ) + Δ ( F ) 覆盖所有距离的至少一半。我们在奇维空间中的结果锐化到一个常数因子。当 E 位于 $${\mathbb {F}}_q^d,$$ F qd 的球面上时，可以看出指数 $$d+\frac{1}{3}$$ d + 1 3改进为 $$d-\frac{1}{6}.$$ d - 1 6 。最后，我们证明了四维向量空间中 Erdős-Falconer 距离猜想的弱版本，用于素域上的乘法子群。