On variants of the Furstenberg set problem

Given an integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d greater-than-or-slanted-equals 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo> ⩾ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d \geqslant 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s element-of left-parenthesis 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>s</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:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">s \in (0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t element-of left-bracket 0 comma 2 left-parenthesis d minus 1 right-parenthesis right-bracket"> <mml:semantics> <mml:mrow> <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>2</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">t \in [0,2(d-1)]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , suppose a set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has the following property: there is a collection of lines of packing dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that every line from the collection intersects <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a set of packing dimension at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that such sets must have packing dimension at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="max left-brace right-brace comma s comma slash slash t 2"> <mml:semantics> <mml:mrow> <mml:mo movablelimits="true" form="prefix">max</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\max \{s,t/2\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and that this bound is sharp. In particular, the special case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d equals 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> solves a variant of the <italic>Furstenberg set problem</italic> for packing dimension. We also solve the upper and lower box dimension variants of the problem. In both of these cases the sharp threshold is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="max left-brace right-brace comma s comma minus minus plus plus t 1 d"> <mml:semantics> <mml:mrow> <mml:mo movablelimits="true" form="prefix">max</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mi>d</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\max \{s,t+1-d\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .