Quantitative rational approximation on spheres

Abstract

We prove a quantitative theorem for Diophantine approximation by rational points on spheres. Our results are valid for arbitrary unimodular lattices and we further prove 'spiraling' results for the direction of approximates. These results are quantitative generalizations of the Khintchine-type theorem on spheres proved in [KM15].

Quantitative Diophantine approximation with congruence conditions

“…The second step is then an argument reducing the counting result on generic lattices in the space of lattices to generic lattices in a much smaller submanifold of positive co-dimension. We note that this was also the strategy used in [APT16,AG20a,AG20b] where similar quantitative results were proved in various settings for the special approximating function ψ 0 (t) = c t with c > 0. The arguments used in [APT16, AG20a, AG20b] for the counting result in step one use ergodic theory and are special to ψ 0 .…”

Effective rational approximation on spheres

“…For rational approximation on spheres, Kleinbock and Merrill [KM13] proved an analog of Dirichlet's theorem and a Kintchine-type dichotomy. For the divergence case in this last result, Alam and Ghosh [AG20] proved a quantitative estimate for the number of rational approximants on spheres with the critical Dirichlet exponent (Theorem 1.1). Our main result in this paper is to prove an estimate with error term analogous to (1.2) for intrinsic Diophantine approximation on the sphere S n .…”

“…Outline of the paper. We develop further the approach of [KM13] and [AG20], starting with the embedding of S n in the positive light cone C := {x ∈ R n+1 × R + : Q(x) = 0} for a quadratic form Q of inertia (n + 1, 1), and identifying good approximants p q ∈ S n with integer points (p, q) in Z n+2 ∩ C whose images under certain rotations k ∈ K = SO(n + 1) lie in a certain domain E T,c ⊂ C (we recall more details about this correspondance in Section 2). The number of solutions N T,c is then related to the number of lattice points in the domain E T,c , which can be appoximated by a more convenient domain F T,c and tessellated by the action of a hyperbolic subgroup {a t , t ∈ R} ⊂ SO(Q).…”

“…In [AG20], the authors use ergodicity of the {a t }-action and Birkoff's Ergodic Theorem to obtain an asymptotic estimate for these ergodic averages. In order to obtain an estimate with error term, we use effective pointwise equidistribution along {a t }-orbits, which we derive from effective double equidistribution of K-orbits pushed by {a t } (Proposition 4.1), using a general method presented in [KSW17] to derive an "almost everywhere"-bound from an L p -bound on ergodic averages (Proposition 4.2).…”

“…We recall the correspondence presented in [KM13] and [AG20] between Diophantine approximation on the sphere S n and the dynamics of orthogonal lattices in R n+2 .…”