Modular hyperbolas and Beatty sequences

Abstract

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

13 pages, 1 figure