Abstract:We investigate the Fourier dimension, Hausdorff dimension, and Lebesgue measure of sets of the form $RY + Z,$ where $R$ is a set of scalars and $Y, Z$ are subsets of Euclidean space. Regarding the Fourier dimension, we prove that for each $\alpha \in [0,1]$ and for each non-empty compact set of scalars $R \subseteq (0,\infty )$, there exists a compact set $Y \subseteq [1,2]$ such that $\dim _F(Y) = \dim _H(Y) = \overline {\dim _M}(Y) = \alpha$ and $\dim _F(RY) \geq \min \{ 1, \dim _F(R) + \dim _F(Y)\}$. This theorem verifies a weak form of a more general conjecture, and it can be used to produce new Salem sets from old ones. Further, we investigate lower bounds on the measure and dimension of $RY+Z$ for $R\subset (0,\infty )$ and arbitrary $Y$ and $Z$; these latter results provide a generalized variant of some theorems of Wolff and Oberlin in which $Y$ is the unit sphere.

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中文翻译：

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总和和乘积的度量和维数

摘要：我们研究了形式为 $RY + Z,$ 的集合的傅立叶维数、Hausdorff 维数和 Lebesgue 测度，其中 $R$ 是一组标量，$Y、Z$ 是欧几里得空间的子集。关于傅立叶维度，我们证明对于每个 $\alpha \in [0,1]$ 和每个非空紧致标量集 $R \subseteq (0,\infty )$，都存在紧致集 $Y \subseteq [1,2]$ 使得 $\dim _F(Y) = \dim _H(Y) = \overline {\dim _M}(Y) = \alpha$ 和 $\dim _F(RY) \geq \ min \{ 1, \dim _F(R) + \dim _F(Y)\}$。这个定理验证了一个更一般的猜想的弱形式，它可以用来从旧的塞勒姆集产生新的塞勒姆集。此外，我们研究了 $R\subset (0,\infty)$ 和任意 $Y$ 和 $Z$ 的 $RY+Z$ 的度量和维度的下限；

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