AI 中文总结
研究自相似测度正交投影的绝对连续性，通过弗斯滕伯格型准则及利用特定拉马努金集，得到奇异自相似测度线投影绝对连续、小傅里叶维数测度在多数方向光滑投影等结果。
AI 中文摘要
我们证明了一个关于自相似测度的给定正交投影为绝对连续且具有量化正则性的弗斯滕伯格型准则。它要求旋转部分以与轨道相对类似维数相比足够快的速率进行指数混合。利用卢博茨基、菲利普斯和萨纳克（1986年、1987年）构造的\(\mathrm{SO}(3)\)中无理旋转的拉马努金集，我们得到了明确的应用。特别地，我们展示了奇异自相似测度，其每条线投影都是绝对连续的；具有任意小傅里叶维数的测度，在除一个完全明确的例外方向集之外的所有方向上都有光滑投影；以及一个具有\(C^2_0\)密度的塞勒姆自相似测度的非平凡例子。
英文摘要
We prove a Furstenberg-type criterion for a given orthogonal projection of a self-similar measure to be absolutely continuous, with quantified regularity. It requires exponential mixing of the rotational part at a rate that is sufficiently fast compared with an orbit relative analogue of its dimension. Using Ramanujan sets of irrational rotations in \(\mathrm{SO}(3)\) constructed by Lubotzky, Phillips and Sarnak (1986, 1987), we obtain explicit applications. In particular, we exhibit singular self-similar measures whose every line projection is absolutely continuous, measures of arbitrarily small Fourier dimension with smooth projections in all but a fully explicit exceptional set of directions, and a non-trivial example of a self-similar measure that is Salem with a $C^2 _0$ density.
