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Simon Kristensen

Publications and source records attributed to Simon Kristensen.

26 records · Page 2Linked to original sources
arXiv2008

On shrinking targets for Z^m actions on tori

Let A be an n by m matrix with real entries. Consider the set Bad_A of x \in [0,1)^n for which there exists a constant c(x)>0 such that for any q \in Z^m the distance between x and the point {Aq} is at least c(x) |q|^{-m/n}. It is shown that the intersection of Bad_A with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.

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arXiv2007

Diophantine exponents for mildly restricted approximation

We are studying the Diophantine exponent μ_{n,l}$ defined for integers 1 \leq l < n and a vector α\in \mathbb{R}^n by letting μ_{n,l} = \sup{μ\geq 0: 0 < ||x \cdot α|| < H(x)^{-μ} for infinitely many x \in C_{n,l} \cap \mathbb{Z}^n}, where \cdot is the scalar product and || . || denotes the distance to the nearest integer and C_{n,l} is the generalised cone consisting of all vectors with the height attained among the first l coordinates. We show that the exponent takes all values in the interval [l+1, \infty), with the value n attained for almost all α. We calculate the Hausdorff dimension of the set of vectors αwith μ_{n,l} (α) = μfor μ\geq n. Finally, letting w_n denote the exponent obtained by removing the restrictions on x, we show that there are vectors αfor which the gaps in the increasing sequence μ_{n,1} (α) \leq ... \leq μ_{n,n-1} (α) \leq w_n (α) can be chosen to be arbitrary.

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arXiv2005

Diophantine approximation and badly approximable sets

Let (X,d) be a metric space and (Ω, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of Ω. Loosely speaking, these consist of points in Ωwhich `stay clear' of some given set of points in X. The classical set \Bad of `badly approximable' numbers in the theory of Diophantine approximation falls within our framework as do the sets \Bad(i,j) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Ωhave full Hausdorff dimension. Applications of our general framework include those from number theory (classical, complex, p-adic and formal power series) and dynamical systems (iterated function schemes, rational maps and Kleinian groups).

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arXiv2004

Badly approximable systems of linear forms over a field of formal series

We prove that the Hausdorff dimension of the set of badly approximable systems of m linear forms in n variables over the field of Laurent series with coefficients from a finite field is maximal. This is a analogue of Schmidt's multi-dimensional generalisation of Jarnik's Theorem on badly approximable numbers.

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arXiv2003

Hausdorff Dimension and Diophantine Approximation

We begin with a brief treatment of Hausdorff measure and Hausdorff dimension. We then explain some of the principal results in Diophantine approximation and the Hausdorff dimension of related sets, originating in the pioneering work of Vojtech Jarnik. We conclude with some applications of these results to the metrical structure of exceptional sets associated with some famous problems. It is not intended that all the recent developments be covered but they can be found in the references cited.

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