Modular hyperbolas and Beatty sequences

Bounds for $\max \{m,\tilde{m}\}$ subject to $m,\tilde{m} \in $\mathbb{Z}$\cap[1,p)$,$p$prime,$z$indivisibleby$p$,$m\tilde{m}\equiv z\bmod p$and$m$belongingtosomefixedBeattysequence${ \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 $0,, \operatorname{e}_p(y\overline{1}), \operatorname{e}_p(y\overline{2}), \ldots, \operatorname{e}_p(y(\overline{p-1}))$onaverageareobtained,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.