关于Furstenberg交猜想、自相似测度和卷积的$L^q$范数
On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions
摘要：
研究了实直线上一类具有自相似结构的测度，我们称之为动态驱动的自相似测度，并将Bernoulli卷积等自相似测度作为其特例.我们的主要结果给出了一个表达式的$L^q$维的动态驱动的自相似措施，在一定的条件下。作为应用，我们解决了Furstenberg关于$\times p$和$\times q$ -不变集的交集维数的猜想.在其他几个应用程序中，我们还表明，伯努利卷积有一个$L^q$密度为所有有限的$q$，一个零维集的例外。主要结果的证明受M. Hochman对自相似测度维数的处理方法和熵的逆定理。我们的方法可以被看作是一个扩展的Hochman的理论从熵$L^q$规范，同样依赖于逆定理的衰减$L^q$规范的离散措施卷积。我们的方法的这一核心部分可能是独立的兴趣，并且是众所周知的方法和结果在加法组合学中的应用：由于Tao-Vu的Balog-Szemerédi-Gowers定理的不对称版本，以及Bourgain的一些构造。
英文摘要：
We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the $L^q$-dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of $\times p$ and $\times q$-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an $L^q$ density for all finite $q$, outside of a zero-dimensional set of exceptions. The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to $L^q$ norms, and likewise relies on an inverse theorem for the decay of $L^q$ norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemerédi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.
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