论文标题
shmerkin的新动态证明 - wu定理
A new dynamical proof of the Shmerkin--Wu theorem
论文作者
论文摘要
令$ a <b $为多重独立整数，都至少$ 2 $。令$ a，b $封闭的子集为$ [0,1] $，分别在乘以$ a $，$ b $的情况下向前不变，让$ c：= a \ times b $。 Furstenberg的一个旧猜想断言，任何与任何一个轴平行的平面线$ L $都必须在Hausdorff尺寸中相交，最多只能在$ \ max \ {\ dim c，1 \} -1 $中相交。 Shmerkin和Wu最近的两幅作品给出了这一猜想的两种不同的证据。该注释提供了第三个证明。像Wu一样，它与Furstenberg介绍的有关研究此类问题的奇异理论机制保持着亲密关系，但它使用了诸如厄运理论的实质背景较少。还使用了相同的方法来重新启动YU的最新结果，涉及某些总和序列。
Let $a < b$ be multiplicatively independent integers, both at least $2$. Let $A,B$ be closed subsets of $[0,1]$ that are forward invariant under multiplication by $a$, $b$ respectively, and let $C := A\times B$. An old conjecture of Furstenberg asserted that any planar line $L$ not parallel to either axis must intersect $C$ in Hausdorff dimension at most $\max\{\dim C,1\} - 1$. Two recent works by Shmerkin and Wu have given two different proofs of this conjecture. This note provides a third proof. Like Wu's, it stays close to the ergodic theoretic machinery that Furstenberg introduced to study such questions, but it uses less substantial background from ergodic theory. The same method is also used to re-prove a recent result of Yu about certain sequences of sums.
