Let $X$ be a symmetric Lévy process in $\mathbb{R}^d, d =
2, 3$. We assume that $X$ has independent $\alpha_j$-stable components, $1 <
\alpha_d \leq \dots \leq \alpha_1 < 2$ (a process with stable components, by
Pruitt and Taylor), or more generally that $X$ is $d$-dimensionally
self-similar with similarity exponents $H_j, H_j = 1/\alpha_j$ (a
dilation-stable process, by Kunita). Let a given integer $k \geq 2$ be such
that $k(H - 1) < H, H = \Sigma_{j=1}^d H_j$. We prove that the set of
$k$-multiple points $E_k$ is almost surely of Hausdorff dimension
$$\dim E_k = \min (\frac{k - (k -1)H}{H_1}, d -
\frac{k(H - 1)}{H_d})$$.
In the stable components case, the above formula was
proved by Hendricks for $d = 2$ and was suspected by him for $d = 3$.