A new dynamical proof of the Shmerkin-Wu theorem (Q2076562)

Suppose \(a\) and \(b\), where \(1<a<b\), are multiplicatively independent integers and \(A,B\) are closed subsets of \([0, 1]\) such that are forward invariant under multiplication by \(a, b\) respectively, and also suppose that \(C:=A \times B\). The present paper is devoted to a new proof of the following theorem named as the Shmerkin-Wu theorem or the Furstenberg conjecture: Theorem. If \(L\) is any line not parallel to either coordinate axis, then \[ \dim_H (C \cap L) \le \max\{\dim_H (C), 1\} -1, \] where \(\dim_H(\cdot)\) is the Hausdorff dimension of a set. The special attention is given to known two proofs of the last theorem which were given by \textit{P. Shmerkin} [Ann. Math. (2) 189, No. 2, 319--391 (2019; Zbl 1426.11079)] and \textit{M. Wu} [ibid. 189, No. 3, 707--751 (2019; Zbl 1430.11106)]. Techniques of these proofs and some applications are briefly considered. Certain auxiliary notions and results of the geometric measure theory, are briefly described. One can note that a generalization of the Marstrand theorem and adic versions of the mass distribution principle are considered. A new proof of Yu's theorem on certain sequences of sums, is given. It is useful for illustrating the approach to the main proof. The third proof of the main theorem is presented with explanations.