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Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are replaced with a countable hierarchy of `well-spread points, which we refer to as abstract rationals. We prove various Jarnik-Besicovitch type dimension bounds and investigate their sharpness.
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A {it two-dimensional continued fraction expansion} is a map $mu$ assigning to every $x inmathbb R^2setminusmathbb Q^2$ a sequence $mu(x)=T_0,T_1,dots$ of triangles $T_n$ with vertices $x_{ni}=(p_{ni}/d_{ni},q_{ni}/d_{ni})inmathbb Q^2, d_{ni}>0, p_{ni}, q_{ni}, d_{ni}in mathbb Z,$ $i=1,2,3$, such that begin{eqnarray*} det left(begin{matrix} p_{n1}& q_{n1} &d_{n1} p_{n2}& q_{n2} &d_{n2} p_{n3}& q_{n3} &d_{n3} end{matrix} right) = pm 1,,, ,,,mbox{and},,,,,, bigcap_n T_n = {x}. end{eqnarray*} We construct a two-dimensional continued fraction expansion $mu^*$ such that for densely many (Turing computable) points $x$ the vertices of the triangles of $mu(x)$ strongly converge to $x$. Strong convergence depends on the value of $lim_{nto infty}frac{sum_{i=1}^3dist(x,x_{ni})}{(2d_{n1}d_{n2}d_{n3})^{-1/2}},$ (dist denoting euclidean distance) which in turn depends on the smallest angle of $T_n$. Our proofs combine a classical theorem of Davenport Mahler in diophantine approximation, with the algorithmic resolution of toric singularities in the equivalent framework of regular fans and their stellar operations.
Recently, Adiceam, Beresnevich, Levesley, Velani and Zorin proved a quantitative version of the convergence case of the Khintchine-Groshev theorem for nondegenerate manifolds, motivated by applications to interference alignment. In the present paper, we obtain analogues of their results for affine subspaces.
We prove the convergence and divergence cases of an inhomogeneous Khintchine-Groshev type theorem for dual approximation restricted to affine subspaces in $mathbb{R} ^n$. The divergence results are proved in the more general context of Hausdorff measures.
In 2004, J.C. Tong found bounds for the approximation quality of a regular continued fraction convergent of a rational number, expressed in bounds for both the previous and next approximation. We sharpen his results with a geometric method and give both sharp upper and lower bounds. We also calculate the asymptotic frequency that these bounds occur.
In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue of Khintchines theorem for these sets. We then apply this result to the topic of intrinsic Diophantine Approximation on self-similar sets. In particular, we define a new height function for an element of $mathbb{Q}^d$ contained in a self-similar set in terms of its eventually periodic representations. For limsup sets defined with respect to this height function, we obtain a detailed description of their metric properties. The results of this paper hold in arbitrary dimensions and without any separation conditions on the underlying iterated function system.
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