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Journal of the London Mathematical Society 1999 60(2):381-398; doi:10.1112/S0024610799007681
© 1999 by London Mathematical Society
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© The London Mathematical Society

The Shrinking Target Problem for Matrix Transformations of Tori

Richard Hill and Sanju L. Velani

Department of Mathematics, University College London Gower Street, London
Department of Mathematics, Queen Mary and Westfield College Mile End Road, London

Received 16 August 1996. Revision received 10 November 1997.

Let T be a dxd matrix with integral coefficients. Then T determines a self-map of the d-dimensional torus X = Rd/Zd. Choose for each natural number n a ball B(n) in X and suppose that B(n+1) has smaller radius than B(n) for all n. Now let W be the set of points x isin X such that Tn(x) isin B(n) for infinitely many n isin N. The Hausdorff dimension of W is studied by analogy with the Jarník–Besicovitch theorem on the dimension of the set of well-approximable real numbers. The dimension depends on the quantity


Formula

A complete description is given only when the matrix is diagonalizable over Q. In other cases a result is obtained for sufficiently large {tau}. The results, in as far as they go, show that the Hausdorff dimension of W is a strictly decreasing, continuous function of {tau} which is piecewise of the form (A{tau}+B)/(C{tau}+D). The numbers A, B, C and D which arise in this way are typically sums of logarithms of the absolute values of eigenvalues of T.


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