AI 中文总结
研究偶数维的埃尔德什 - 法尔科纳距离问题，通过证明提取定理，将偶数维猜想归结为平面情形，在固定距离问题和三角形分布上得到改进阈值，在素域和任意有限域分别取得新记录。
AI 中文摘要
设\(q\)为奇素数幂，\(\mathbb{F}_q\)为阶为\(q\)的有限域。我们证明了偶数维埃尔德什 - 法尔科纳距离猜想的一个提取定理，表明所有偶数维的猜想都归结为平面情形。作为推论，我们在固定距离问题和三角形分布上得到了改进的阈值，分别在素域上达到了\(\frac{d}{2}+\frac{1}{4}\)和在任意有限域上达到了\(\frac{d + 1}{2}+\frac{1}{10}\)的新记录。
英文摘要
Let $q$ be an odd prime power and $\mathbb{F}_q$ be the finite field of order $q$. We prove an extraction theorem for the Erdős-Falconer distance conjecture in even dimensions, showing that the conjecture for all even dimensions reduces to the planar case. As consequences, we obtain improved thresholds on the pinned distance problem and the distribution of triangles, achieving new records of $\frac{d}{2}+\frac{1}{4}$ over prime fields and $\frac{d+1}{2}+\frac{1}{10}$ over arbitrary finite fields, respectively.
Comments18 pages
