Extrinsic Diophantine approximation on manifolds and fractals

Fix $d\in\mathbb N$, and let $S\subseteq\mathbb R^d$ be either a real-analytic manifold or the limit set of an iterated function system (for example, $S$ could be the Cantor set or the von Koch snowflake). An $extrinsic$ Diophantine approximation to a point $\mathbf x\in S$ is a rational point $\mathbf p/q$ close to $\mathbf x$ which lies $outside$ of $S$. These approximations correspond to a question asked by K. Mahler ('84) regarding the Cantor set. Our main result is an extrinsic analogue of Dirichlet's theorem. Specifically, we prove that if $S$ does not contain a line segment, then for every $\mathbf x\in S\setminus\mathbb Q^d$, there exists $C > 0$ such that infinitely many vectors $\mathbf p/q\in \mathbb Q^d\setminus S$ satisfy $\|\mathbf x - \mathbf p/q\| < C/q^{(d + 1)/d}$. As this formula agrees with Dirichlet's theorem in $\mathbb R^d$ up to a multiplicative constant, one concludes that the set of rational approximants to points in $S$ which lie outside of $S$ is large. Furthermore, we deduce extrinsic analogues of the Jarn\'ik--Schmidt and Khinchin theorems from known results

Unconventional height functions in simultaneous Diophantine approximation

Simultaneous Diophantine approximation is concerned with the approximation of a point $\mathbf x\in\mathbb R^d$ by points $\mathbf r\in\mathbb Q^d$, with a view towards jointly minimizing the quantities $\|\mathbf x - \mathbf r\|$ and $H(\mathbf r)$. Here $H(\mathbf r)$ is the so-called "standard height" of the rational point $\mathbf r$. In this paper the authors ask: What changes if we replace the standard height function by a different one? As it turns out, this change leads to dramatic differences from the classical theory and requires the development of new methods. We discuss three examples of nonstandard height functions, computing their exponents of irrationality as well as giving more precise results. A list of open questions is also given

Intrinsic Approximation on Cantor-like Sets, a Problem of Mahler

In 1984, Kurt Mahler posed the following fundamental question: How well can irrationals in the Cantor set be approximated by rationals in the Cantor set? Towards development of such a theory, we prove a Dirichlet-type theorem for this intrinsic diophantine approximation on Cantor-like sets, and discuss related possible theorems/conjectures. The resulting approximation function is analogous to that for R^d, but with d being the Hausdorff dimension of the set, and logarithmic dependence on the denominator instead.Comment: 7 pages, 0 figure

Diophantine approximation in Banach spaces

In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points