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arXiv · 2608.15458

Five-Term and Higher Congruences Involving Arbitrary Sets and Short Intervals Modulo a Prime

Yao Zhi
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Abstract

We obtain asymptotic formulas for additive congruences \[ \sum_{i=1}^r m_i x_i^{-s}\equiv \lambda \pmod p, \] 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 \(\lambda\) 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 \[ \alpha_r=\frac13+\frac{4}{9\sqrt r}+O(r^{-1}) \] as \(r\to\infty\).

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Yao Zhi. 2026-08-16. Five-Term and Higher Congruences Involving Arbitrary Sets and Short Intervals Modulo a Prime. https://arxiv.org/abs/2608.15458

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