A QUANTITATIVE DYNAMICAL BOREL–CANTELLI LEMMA FOR TWISTED RECURRENCE ON [ 0 , 1 ] d [0,1]ᵈleft bracket 0 comma 1 right bracket Superscript d

Let T : [ 0 , 1 ] d → [ 0 , 1 ] d T:[0,1]ᵈ → [0,1]ᵈ upper T colon left bracket 0 comma 1 right bracket Superscript d Baseline right arrow left bracket 0 comma 1 right bracket Superscript d be a piecewise expanding linear map such that each domain of linearity is a coordinate-parallel hyperrectangle that T sends to [ 0 , 1 ] d [0,1]ᵈ left bracket 0 comma 1 right bracket Superscript d . Let { r n } ₙ\ StartSet bold r Subscript n Baseline EndSet be a sequence of vectors with r n = ( r n , 1 , … , r n , d ) ∈ ( R ≥ 0 ) d rₙ=(rn, 1, … , rn, d) ∈ (R≥ 0)ᵈ bold r Subscript n Baseline equals left parenthesis r Subscript n comma 1 Baseline comma ellipsis comma r Subscript n comma d Baseline right parenthesis element of left parenthesis double struck upper R Subscript greater than or equals 0 Baseline right parenthesis Superscript d . Let m d md m Subscript d denote d -dimensional Lebesgue measure. Let <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.or