涉及素数模任意集合与短区间的五项及更高项同余式
Five-Term and Higher Congruences Involving Arbitrary Sets and Short Intervals Modulo a Prime
AI总结：
该研究针对素数模的五项及更高项加法同余式，利用双指数和的中心四阶矩估计，推导了亚平方根范围的渐近公式，为同余式问题提供了新的阈值结果。
AI中文摘要：
我们针对加法同余式\\(\sum_{i=1}^r m_i x_i^{-s}\equiv \lambda \pmod p\\)，其中\\(m_i\\)取自\\(\mathbb F_p^\ast\\)的任意子集，\\(x_i\\)取自移位区间，推导了渐近公式。对于五项且基数均为\\(N\\)的平衡情形，当\\(N>p^{14/29+\varepsilon}\\)时，该渐近式对所有\\(\lambda\\)一致成立，达到了真正的亚平方根范围。主要输入是关联双指数和的中心四阶矩估计，相同方法对每个固定\\(r\ge5\\)均能得到亚平方根阈值，包括六项时\\(N>p^{8/17+\varepsilon}\\)，且当\\(r\to\infty\\)时\\(\alpha_r=\frac13+\frac{4}{9\sqrt r}+O(r^{-1})\\)。
英文摘要：
We obtain asymptotic formulas for additive congruences \[ \sum_{i=1}^r m_i x_i^{-s}\equiv λ\pmod p, \] where the \(m_i\) range over arbitrary subsets of \(\mathbb F_p^\ast\) and the \(x_i\) over shifted intervals. For five terms, in the balanced case of common cardinality \(N\), the asymptotic holds uniformly in \(λ\) whenever \[ N>p^{14/29+\varepsilon}, \] giving a genuine sub-square-root range. The main input is a centered fourth-moment estimate for the associated double exponential sums. The same method yields sub-square-root thresholds for every fixed \(r\ge 5\), including \(N>p^{8/17+\varepsilon}\) for six terms, with \[ α_r=\frac13+\frac{4}{9\sqrt r}+O(r^{-1}) \] as \(r\to\infty\).
