2022/08/12 by Bing Li, Lingmin Liao, Li, Bing +6 · 8 citations
Mathematics · #11J83 (Primary) #28A78 #28A80 #37E05 (Secondary) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2208.06112
openalex publication_date 2022/08/12 · openalex created_date 2022/08/16 · openalex updated_date 2026/07/28
Let T be a d× d matrix with real coefficients. Then T determines a self-map of the d-dimensional torus \Bbb Td=ℝd/\Bbb Zd. Let \En \n ∈ ℕ be a sequence of subsets of \Bbb Td and let W(T,\En \) be the set of points x ∈ \Bbb Td such that Tn(x)∈ En for infinitely many n∈ ℕ. For a large class of subsets (namely, those satisfying the so called bounded property (\boldsymbol\rm B) which includes balls, rectangles, and hyperboloids) we show that the d-dimensional Lebesgue measure of the shrinking target set W(T,\En \) is zero (resp. one) if a natural volume sum converges (resp. diverges). In fact, we prove a quantitative form of this zero-one criteria that describes the asymptotic behaviour of the counting function R(x,N):= # \ 1≤ n ≤ N : Tn(x) ∈ En \ . The counting result makes use of a general quantitative statement that holds for a large class measure-preserving dynamical systems (namely, those satisfying the so called summable-mixing property). We next turn our attention to the Hausdorff dimension of W(T,\En \). In the case the subsets En are balls, rectangles or hyperboloids we obtain precise formulae for the dimension. These shapes correspond, respectively, to the simultaneous, weighted and multiplicative theories of classical Diophantine approximation. The dimension results for balls generalises those obtained in an earlier paper by Hill and the third-named author for integer matrices to real matrices. In the final section, we discuss various problems that stem from the results proved in the paper.