A QUANTITATIVE DYNAMICAL BOREL–CANTELLI LEMMA FOR TWISTED RECURRENCE ON <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/

Abstract Let <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>T</mml:mi> <mml:mo>:</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy="false">→</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mi>d</mml:mi> </mml:msup> </mml:math> $T:[0,1]^d \rightarrow [0,1]^d$ 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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mi>d</mml:mi> </mml:msup> </mml:math> $[0,1]^d$ left bracket 0 comma 1 right bracket Superscript d . Let <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mo stretchy="false" fence="false">{</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="bold">r</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false" fence="false">}</mml:mo> </mml:math> $\{\mathbf {r}_n\}$ StartSet bold r Subscript n Baseline EndSet be a sequence of vectors with <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mrow> <mml:mi mathvariant="bold">r</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>d</mml:mi> </mml:msup> </mml:math> $\mathbf {r}_n=(r_{n, 1}, \ldots , r_{n, d}) \in (\mathbb {R}_{\geq 0})^d$ 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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> $m_d$ 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