论文标题
Furstenberg套装的尺寸和Bourgain投影定理的扩展
Dimensions of Furstenberg sets and an extension of Bourgain's projection theorem
论文作者
论文摘要
我们表明，$（s，t）$ - furstenberg集的Hausdorff尺寸至少为$ s+t/2+ε$，其中$ε> 0 $仅取决于$ s $和$ t $。这改善了以前最著名的限制，价格为$ 2S <t \ le 1+ε（s，t）$，尤其是自1999年以来的首次改进到经典$ s $ s $ -s -furstenberg套件的尺寸，价格为$ s <1/2 $。我们从最低限度的非集中假设下的相应离散的发病率中推断出来，该假设同时扩展了波尔加因的离散投影和总成果定理。这些证明是基于t。〜orponen的最新离散发病率和第一作者，以及$（s，t）$和$（t/2，s+t/2）$ - furstenberg set之间的一定双重性。
We show that the Hausdorff dimension of $(s,t)$-Furstenberg sets is at least $s+t/2+ε$, where $ε>0$ depends only on $s$ and $t$. This improves the previously best known bound for $2s<t\le 1+ε(s,t)$, in particular providing the first improvement since 1999 to the dimension of classical $s$-Furstenberg sets for $s<1/2$. We deduce this from a corresponding discretized incidence bound under minimal non-concentration assumptions, that simultaneously extends Bourgain's discretized projection and sum-product theorems. The proofs are based on a recent discretized incidence bound of T.~Orponen and the first author and a certain duality between $(s,t)$ and $(t/2,s+t/2)$-Furstenberg sets.
