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The Shrinking Target Problem for Matrix Transformations of Tori

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JournalJournal of the London Mathematical Society
DatePublished - Oct 1999
Issue number2
Volume60
Number of pages18
Pages (from-to)381-398
Original languageEnglish

Abstract

Let T be a d×d 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 ¿ X such that Tn(x) ¿ B(n) for infinitely many n ¿ 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 t. The results, in as far as they go, show that the Hausdorff dimension of W is a strictly decreasing, continuous function of t which is piecewise of the form (At+B)/(Ct+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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