Approximation of points in the plane by generic lattice orbits

Abstract

We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice Γ < SL 2 (R) acting linearly on R 2 . Our method gives bounds that are uniform for almost all orbits.

Shrinking targets for semisimple groups

“…In , we used it to prove a finer result regarding the density of values taken by generic indefinite forms. In , the second author used this method to study the rate of approximation by generic lattice orbits acting on the plane, and in a forthcoming paper by Kelmer, he used it to study shrinking target problems for unipotent flows on homogeneous spaces.…”

Diophantine approximation with nonsingular integral transformations

“…Moreover, for x for which h(x) ≥ 7 5 the value of ψ as in the conclusion exceeds 3, whereas existence of solutions is assured with ψ = 3 by the result in [9] for the action of SL(2, Z) and hence that of Γ. Thus for x with h(x) ≥ 7 5 , [9] offers better results; however the set of x for which that happens has measure 0.…”

“…There has been considerable interest in the literature in effective results of this kind, for various group actions. In particular it was shown in [9], for n = 2, that given an irrational vector x in R 2 and any target vector y ∈ R 2 there exist a constant C = C(x, y) and infinitely many γ in SL(2, Z) such that ||γx − y|| ≤ C||γ|| − 1 3 ; there are also stronger results proved in [9] under some restrictions on y, which we shall not go into here; see also [4], [5], [7], [8], [11], and [12], for analogous results for various actions; it may be mentioned that while these works address this question of exponents in many contexts, they however do not cover the setup that is dealt within this paper. Here we describe some results along this theme for the action of Γ as above; for the case n = 2 the result is stronger in import than the result recalled above for SL(2, Z), in the sense that for almost all initial points x ∈ R 2 the corresponding statement holds for all ρ less than 1, in place of ρ = 1 3 for SL(2, Z); see also Remark 4.3.…”

Diophantine approximation with nonsingular integral transformations

“…proved in [7] under some restrictions on y, which we shall not go into here; see also [9], [6] and [4] for analogous results for various actions; it may be mentioned that these results are broader in their framework, but weaker in terms of the exponents involved.…”