Bounds for $\max\{m,\tilde{m}\}$ subject to $m,\tilde{m} \in \mathbb{Z}\cap[1,p)$, $p$ prime, $z$ indivisible by $p$, $m\tilde{m}\equiv z\bmod p$ and $m$ belonging to some fixed Beatty sequence $\{ \lfloor n\alpha+\beta \rfloor : n\in\mathbb{N} \}$ are obtained, assuming certain conditions on $\alpha$. The proof uses a method due to Banks and Shparlinski. As an intermediate step, bounds for the discrete periodic autocorrelation of the finite sequence $0,\, \operatorname{e}_p(y\overline{1}), \operatorname{e}_p(y\overline{2}), \ldots, \operatorname{e}_p(y(\overline{p-1}))$ on average are obtained, where $\operatorname{e}_p(t) = \exp(2\pi i t/p)$ and $m\overline{m} \equiv 1\bmod p$. The latter is accomplished by adapting a method due to Kloosterman.

中文翻译：

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模双曲线和比蒂序列

$\max\{m,\tilde{m}\}$ 的边界受 $m,\tilde{m} \in \mathbb{Z}\cap[1,p)$, $p$ prime, $z $ 不能被 $p$、$m\tilde{m}\equiv z\bmod p$ 和属于某个固定比蒂序列的 $m$ 整除 $\{ \lfloor n\alpha+\beta \rfloor : n\in\mathbb{ N} \}$ 得到，假设 $\alpha$ 上的某些条件。证明使用了 Banks 和 Shparlinski 的方法。作为中间步骤，有限序列 $0,\, \operatorname{e}_p(y\overline{1}), \operatorname{e}_p(y\overline{2}), \operatorname{e}_p(y\overline{2}), \operatorname{e}_p(y\overline{2}), \ ldots, \operatorname{e}_p(y(\overline{p-1}))$ 平均得到，其中 $\operatorname{e}_p(t) = \exp(2\pi it/p)$ 和$m\overline{m} \equiv 1\bmod p$。后者是通过采用 Kloosterman 的方法来实现的。