Thick points for intersections of planar sample paths

Let $L_n^{X}(x)$ denote the number of visits to $x \in {\bf Z}^2$ of the simple planar random walk $X$, up till step $n$. Let $X'$ be another simple planar random walk independent of $X$. We show that for any $0<b<1/(2 \pi)$, there are $n^{1-2\pi b+o(1)}$ points $x \in {\bf Z}^2$ for which $L_n^{X}(x)L_n^{X'}(x)\geq b^2 (\log n)^4$. This is the discrete counterpart of our main result, that for any $a<1$, the Hausdorff dimension of the set of {\it thick intersection points} $x$ for which $\limsup_{r \to 0} {\mathcal I}(x,r)/(r^2|\log r|^4)=a^2$, is almost surely $2-2a$. Here ${\mathcal I}(x,r)$ is the projected intersection local time measure of the disc of radius $r$ centered at $x$ for two independent planar Brownian motions run till time 1. The proofs rely on a `multi-scale refinement' of the second moment method. In addition, we also consider analogous problems where we replace one of the Brownian motions by a transient stable process, or replace the disc of radius $r$ centered at $x$ by $x+rK$ for general sets $K$.