Values of random polynomials in shrinking targets

Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $

“…Hence one can apply Proposition 3 for almost all x ∈ G/Γ. We also remark that the set of functions of the form J f ν, where f = f p , ν = ν p with (13) and (14) respectively, is a generating set of , ζ), ∀u ∈ Z p − pZ p , ∀p ∈ S f }. Hence Proposition 3 holds for functions in L as well (See details in [13]).…”

“…Let q S be an isotropic quadratic form of rank n = 3 or 4. Let f = f p and ν = ν p be as in (13) and (14). Assume further that f satisfies the following condition: there is a nonnegative continuous function f + of compact support on Q n S such that supp(f ) ⊂ supp(f + ) • , where A • is an interior of A and sup t 0 K f + (a t kgΓ)dm(k) = M < ∞.…”

“…Ghosh and Kelmer [11] showed another version of the quantitative version of Oppenheim problem for real generic ternary quadratic forms and Ghosh, Gorodnik and Nevo [10] extended this result for more general setting, such as for generic characteristic polynomial maps. Recently, there have been effective results about the bound of the error terms N (a,b),Ω (T ) − V (a,b),Ω (T ) for almost all real quadratic forms of rank at least 3 by Athreya and Margulis [2], for almost all real homogeneous forms of even degree and of rank at least 3 by Kelmer and Yu [14], and for almost all systems of a homogeneous polynomial of even degree and a linear map with the certain condition by Bandi, Ghosh and the author [3].…”

Values of Inhomogeneous Forms at S-integral points

“…Or alternatively, one could fix either the form or the shift and allow the other to vary. In Theorem 1.1 [14], Ghosh Kelmer and Yu noted that an effective result where both form and shift are allowed to vary, follows from an affine analogue of Rogers second moment formula for the space of affine lattices in conjunction with methods from an earlier paper [19] of Kelmer and Yu (which is discussed below). The case where either the form or the shift is fixed is significantly more difficult.…”

“…In [13], Ghosh, Gorodnik and Nevo used effective mean ergodic theorems to prove a variety of results of this flavour including the case of ternary quadratic forms, namely the classical Oppenheim conjecture. The idea of using Rogers' mean value formula to study effective versions of Oppenheim's conjecture is due to Athreya and Margulis [3] and was further developed by Kelmer and Yu [19]. We also mention the work of Bourgain [7] on certain 'uniform' versions of Oppenheim's conjecture for diagonal forms.…”