The Fame and Confession of the Fraternity of R: C: Commonly, of the Rosy Cross

Hausdorff theory of dual approximation on planar curves

“…instead of the set A n (ψ), and the measure-theoretic property of the intersection S n (ψ) ∩ M. One can then formulate a complete analogue of Conjecture 4, namely the Hausdorff theory for simultaneous approximation on manifolds (see [8,Conjecture 8.2] for the precise statement). This analogue of Conjecture 4 for all non-degenerate planar curves has been remarkably established by Vaughan and Velani [32] for the convergence case, and Beresnevich, Dickinson and Velani [11] for the divergence case. Moreover, the divergence case of this analogous conjecture for all non-degenerate analytic manifolds has been subsequently solved by Beresnevich [8].…”

“…Despite the superficial similarity between the simultaneous and dual approximations on manifolds, the two problems exhibit quite differing natures and difficulties; hence the solution of one of them unfortunately does not yield that of the other one by any reasonable means. Our main theorem below is a natural counterpart of the result of Vaughan and Velani [32] in the dual setting, and hence brings the development of the dual approximation on manifolds in line with that of the simultaneous approximation on manifolds.…”

“…The novel feature of this paper is that we adapt a counting result of the same type due to Huxley [25], which is essentially best possible, into an analytic form that is particularly suited for the application in Schmidt's method. This connection is established through the use of the dual curve, which appears, in one form or another, in the previous works [1,11,15,23,25,32].…”

“…Actually, we still obtain results of the same quality when the upper bound in Lemma 4 is relaxed to be ≪ δ α R β + R 3/2−ε for some ε > 0 and α, β satisfying β < 3 2 + 2α. In contrast, we recall that when the corresponding Jarník type theorem for simultaneous approximation on planar curves was established in [32] one had to use Huxley in the strongest form. This discrepancy is probably not surprising since it is generally believed that dual approximation on manifolds is not more difficult than simultaneous approximation on manifolds.…”

“…This discrepancy is probably not surprising since it is generally believed that dual approximation on manifolds is not more difficult than simultaneous approximation on manifolds. Nevertheless, it is indeed a bit surprising that the dual case for planar curves remained open for almost a decade after the simultaneous case was solved completely [11,32]!…”