An improvement on Furstenberg’s intersection problem

In this paper, we study a problem posed by Furstenberg on intersections between × 2 , 3 \times 2, \times 3 invariant sets. We present here direct geometrical counting argument to revisit theorem of Wu and Shmerkin. This can be used obtain further improvements. For example, show that if alttext="upper A upper 3 subset-of left-bracket 0 1 right-bracket"> A ⊂<!-- ⊂ <mml:mo stretchy="false">[ 0 1 stretchy="false">] encoding="application/x-tex">A_2,A_3\subset [0,1] are closed respectively, assuming alttext="dimension plus dimension greater-than 1"> dim ⁡<!-- ⁡ <mml:mo>+ &gt; encoding="application/x-tex">\dim A_2+\dim A_3&gt;1 then intersection left-parenthesis u v right-parenthesis"> ∩<!-- ∩ stretchy="false">( u v stretchy="false">) encoding="application/x-tex">A_2\cap (uA_3+v) is sparse (defined in paper) has box zero uniformly with respect the real parameters alttext="u v"> encoding="application/x-tex">u,v such alttext="u"> encoding="application/x-tex">u Superscript negative −<!-- − </mml:msup> encoding="application/x-tex">u^{-1} both bounded away from alttext="0"> encoding="application/x-tex">0 .