  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.
