标题： 傅里叶衰减和绝对连续性对于 ${\mathbb R}^d$ 中的典型齐次自相似测度在 $d\ge 3$ 显示英文标题

标题： Fourier decay and absolute continuity for typical homogeneous self-similar measures in ${\mathbb R}^d$ for $d\ge 3$

摘要： 我们考虑在${\mathbb R}^d$中的迭代函数系统 (IFS)，对于形式为$\{f_j(x) = \lambda {\mathcal O} x + a_j\}_{j=0}^m$的$d\ge 3$，其中$a_0=0$和$m\ge 1$。 此处$\lambda\in (0,1)$是压缩比，${\mathcal O}$是正交矩阵。 给定一个正概率向量$p$，对于IFS存在一个唯一的不变（平稳）测度，称为（在这种情况下）齐次自相似测度，我们将其表示为$\mu(\lambda {\mathcal O}, {\mathcal D}, p)$，其中${\mathcal D} = \{a_0,\ldots,a_m\}$是“向量数字”的集合。 我们得到了关于此类测度的傅里叶衰减的两个结果。 首先我们证明，如果 ${\mathcal D}$ 张开 ${\mathbb R}^d$，那么对于每个固定的 ${\mathcal O}$ 和 $p$，测度 $\mu(\lambda {\mathcal O}, {\mathcal D}, p)$ 对于 $\lambda$ 的零Hausdorff维集以外的所有情况都具有幂级数傅里叶衰减（等价地，正的傅里叶维数）。 在我们的第二个结果中，我们对${\mathcal D}$不施加任何限制，除了仿射不可约性的必要条件，并得到几乎所有齐次自相似测度的幂傅里叶衰减；然而，仅限于偶数$d\ge 4$。 结合 Corso 和 Shmerkin [arXiv:2409.04608] 的最新工作，这些结果意味着在相同的假设下，在超临界参数区域内，几乎所有自相似测度都是绝对连续的。 显示英文摘要

摘要： We consider iterated function systems (IFS) in ${\mathbb R}^d$ for $d\ge 3$ of the form $\{f_j(x) = \lambda {\mathcal O} x + a_j\}_{j=0}^m$, with $a_0=0$ and $m\ge 1$. Here $\lambda\in (0,1)$ is the contraction ratio and ${\mathcal O}$ is an orthogonal matrix. Given a positive probability vector $p$, there is a unique invariant (stationary) measure for the IFS, called (in this case) a homogeneous self-similar measure, which we denote $\mu(\lambda {\mathcal O}, {\mathcal D}, p)$, where ${\mathcal D} = \{a_0,\ldots,a_m\}$ is the set of ``vector digits''. We obtain two results on Fourier decay for such measures. First we show that if ${\mathcal D}$ spans ${\mathbb R}^d$, then for every fixed ${\mathcal O}$ and $p$ the measure $\mu(\lambda {\mathcal O}, {\mathcal D}, p)$ has power Fourier decay (equivalently, positive Fourier dimension) for all but a zero-Hausdorff dimension set of $\lambda$. In our second result we do not impose any restrictions on ${\mathcal D}$, other than the necessary one of affine irreducibility, and obtain power Fourier decay for almost all homogeneous self-similar measures; however, only for even $d\ge 4$. Combined with recent work of Corso and Shmerkin [arXiv:2409.04608] , these results imply absolute continuity for almost all self-similar measures under the same assumptions, in the super-critical parameter region.