Effective estimates on indefinite ternary forms

Abstract

Abstract. We give an effective proof of a theorem of Dani and Margulis regarding values of indefinite ternary quadratic forms at primitive integer vectors. The proof uses an effective density-type result for orbits of the groups SO(2, 1) on SL(3, R)/ SL(3, Z).

Shrinking targets for semisimple groups

“…One of the main difficulties for establishing effective results is distinguishing between rational forms and irrational forms that are very well approximated by rational ones. We mention the work of Lindenstrauss‐Margulis on this problem, implying that, unless is very well approximated by a rational form, the values of with integer vectors can be as small as for some (see also for other effective results on this problem).…”

An effective Ratner equidistribution result for SL(2,R)⋉R2

“…Moreover, Einsiedler, Margulis and Venkatesh [5] have proved effective equidistribution for large closed orbits of semisimple groups on homogeneous spaces; see also Mohammadi [34] for a more explicit result in the special case of closed SO(2, 1)-orbits in SL(3, Z)\ SL (3, R). Recently also Lindenstrauss and Margulis [25] have obtained an effective density-type result for arbitrary SO(2, 1)-orbits in SL(3, Z)\ SL(3, R), and used this to give an effective proof of a theorem of Dani and Margulis regarding the values of indefinite ternary quadratic forms at primitive integer vectors.…”

An effective equidistribution result for SL(2,R)⋉(R2)⊕k and application to inhomogeneous quadratic forms

“…This was proved in celebrated work by Margulis [21]. An effective version of this result has more recently been obtained by Lindenstrauss and Margulis, [20]. A quantitative (but non‐effective) version of the Oppenheim conjecture for forms of signature with and was proved by Eskin, Margulis and Mozes, [7], and extended to forms of signature (2,2) subject to a Diophantine condition in [8].…”

“…The condition in (20) is very standard in the Diophantine approximation literature; however we are not aware of any discussion of the more general condition in (21). When (20) holds, we will say that ξ is [κ; c]-LFD, and similarly when (21) holds, we will say that ξ is [(κ, α); c]-LFD. Note that being [κ; c]-LFD is equivalent to being [(κ, α); c]-LFD for any α ≥ κ.…”

“…In recent years, there has been an increased interest in obtaining effective versions of Ratner's results in special cases, that is, to provide an explicit rate of density or equidistribution for the orbits of a (non‐horospherical) unipotent flow (cf. [3, 6, 13, 20, 27, 30, 38]). In particular, in [3, 38], effective equidistribution results were obtained for orbits of a 1‐parameter unipotent flow on , using Fourier analysis and methods of from analytic number theory, and in the very recent paper [30], building on similar methods, effective equidistribution of diagonal translates of certain orbits in was established.…”