Hausdorff theory of dual approximation on planar curves

Abstract

Ten years ago, Beresnevich-Dickinson-Velani [10] initiated a project that develops the general Hausdorff measure theory of dual approximation on non-degenerate manifolds. In particular, they established the divergence part of the theory based on their general ubiquity framework. However, the convergence counterpart of the project remains wide open and represents a major challenging question in the subject. Until recently, it was not even known for any single non-degenerate manifold. In this paper, we settle this problem for all curves in R 2 , which represents the first complete theory of its kind for a general class of manifolds.

An Inhomogeneous Jarník type theorem for planar curves

“…The above theorem brings the theory of dual approximation on non-degenerate curves in line with the advances in simultaneous Diophantine approximation on nondegenerate planer curves in [16], and also generalises the result of Huang [24] concerning dual Diophantine approximation on non-degenerate planer curves to the functional inhomogeneous setting. In doing so, we provide a complete solution to Problem 1 in the special case 'M = C, n = 2'.…”

A Jarník-type theorem for a problem of approximation by cubic polynomials

“…Our proof carries through as-is for any other ψ that is multiplicative, but this is by no means the general case. For the case of quadratic polynomials, Hussain [6] and Huang [5] resorted to imposing a fairly restrictive condition on the dimension function and, while this looks artificial, in private correspondence Hussain confirmed that the techniques used in those papers don't allow to remove it.…”

An equivalence principle between polynomial and simultaneous Diophantine approximation

“…By definition we may choose α arbitrarily close to w for certain arbitrarily large m, and to w for all large m, respectively. The claims (20) and (21) follow. Finally we show θ(ζ) ≤ τ (ζ) −1 to settle (22).…”

“…The respective left inequalities in(20) and(21) and θ(ζ) ≥ τ (ζ) −1 follow immediately from Theorem 2.3 and (71). Concerning the respective right inequalities, the estimates w * (ζ) ≤ w(ζ) ≤ w(ζ) and w * (ζ) ≤ w(ζ) ≤ w(ζ) are an easy consequence of…”

“…Generalizing [39], [9] to non-degenerate manifolds, and in other subtle ways, is an active topic in modern metric Diophantine approximation. However, this is not a major concern of this paper and we only refer to [20] for a recent, general result dealing with planar curves. By (5) the sets of U-numbers and T -numbers in fact both have dimension 0.…”