An Inhomogeneous Jarník type theorem for planar curves

Abstract

In metric Diophantine approximation there are two main types of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarník are fundamental in these settings. Recently, there has been substantial progress towards establishing a metric theory of Diophantine approximations on manifolds. In particular, both the Khintchine and Jarník type results have been established for planar curves except for only one case. In this paper, we prove an inhomogeneous Jarník type theorem for convergence on planar curves and in so doing complete the metric theory for both the homogeneous and inhomogeneous settings for approximation on planar curves.

A Fourier-analytic approach to inhomogeneous Diophantine approximation

“…Some general historical remarks. One of the most famous result was first proved by Khintchine and generalized by Groshev, see for example [2,Theorem 1].…”

Diophantine approximation on curves

“…Since for given q 1 there are only ≪ q 2 1 choices for the pair p, q 2 , the sum with (q 1 , q 2 ) restricted to Θ 1 can be estimated as p∈Z,(q1,q2)∈Θ1 g(µ(q 1 , q 2 , p)) ≪ q1∈Z,q1 =0 g (ψ(q 1 )/|q 1 |) q 2 1 , which converges by assumption. We now treat the more delicate sum where (q 1 , q 2 ) ∈ Θ 2 .…”

“…In 2013, the first ever inhomogeneous result regarding the Hausdorff measure (H g -measure) of D n (Ψ) ∩ M for divergence was established in [1] but only for the dimension function g(r) = r s and with a certain convexity condition on the multivariable approximating function Ψ. In 2017, Badziahin-Harrap-Hussain [2] extended Huang's result to the inhomogeneous setting but still within the framework of a single-variable approximating function. Very recently, authors of this paper proved the GBSP for hypersurfaces i.e.…”

A Dichotomy Law for the Diophantine Properties in ‐dynamical Systems

“…More precisely, the spotlight is on the size of the set W y (T, Ψ) := {x ∈ X : |T n x − y| < Ψ(n) for infinitely many n} , where Ψ : N → R >0 is a positive function such that Ψ(n) → 0 as n → ∞. The set W y (T, Ψ) is the dynamical analogue of the classical well-approximable set (e.g., see [1,2,11,22]) and it has close connections to classic Diophantine approximation, for example when T is an irrational rotation or Gauss transformation. It has been an object of significant interest since the pioneering works of Philipp [28] on the µ-measure of W y (T, Ψ) and Hill and Velani [17] on the Hausdorff dimension of W y (T, Ψ).…”