The shrinking target problem for matrix transformations of tori: Revisiting the standard problem

Let T be a d×d matrix with real coefficients. Then determines self-map of the d-dimensional torus Td=Rd/Zd. {En}n∈N sequence subsets Td and let W(T,{En}) set points x∈Td such that Tn(x)∈En for infinitely many n∈N. For large class (namely, those satisfying so called bounded property (B) which includes balls, rectangles, hyperboloids) we show Lebesgue measure shrinking target is zero (resp. one) if natural volume sum converges diverges). In fact, prove quantitative form this zero-one criteria describes asymptotic behaviour counting function R(x,N):=#{1≤n≤N:Tn(x)∈En}. The result makes use general statement holds measure-preserving dynamical systems summable-mixing property). We next turn our attention to Hausdorff dimension W(T,{En}). case En are rectangles or hyperboloids obtain precise formulae dimension. These shapes correspond, respectively, simultaneous, weighted multiplicative theories classical Diophantine approximation. results balls generalises obtained in [25] integer matrices matrices. final section, discuss various problems stem from proved paper.