Values of Inhomogeneous Forms at S-integral points

Abstract

We prove effective versions of Oppenheim's conjecture for generic inhomogeneous forms in the S-arithmetic setting. We prove an effective result for fixed rational shifts and generic forms and we also prove a result where both the quadratic form and the shift are allowed to vary. In order to do so, we prove analogues of Rogers' moment formulae for S-arithmetic congruence quotients as well as for the space of affine lattices. We believe the latter results to be of independent interest.

Some new results in quantitative Diophantine approximation

“…There has been extensive work recently in proving effective results for generic (i.e. a full measure set in the corresponding moduli space) forms, see [1,12,13,22].…”