Inhomogeneous approximation with coprime integers and lattice orbits

Let (ξ, y) be a point in R and ψ : N→ R a function. We investigate the problem of the existence of infinitely many pairs p, q of coprime integers such that |qξ + p− y| ≤ ψ(|q|). We give both unconditional results which are valid for every real pair (ξ, y) with ξ irrational, and metrical results valid for almost all points (ξ, y). We link the subject with density exponents of lattice orbits in R.