Let $A$ be the subset of $\mathbb{R}^n$ defined by $A=\{x\in\mathbb{R}^{n}:|x_{1}-x_{n}|+\sum_{i=1}^{n-1}|x_{i+1}-x_{i}|\leq\ell\}$ for a given $\ell$. Next, sample a point $p$ uniformly in the unit cube, and let $B$ be the $\ell_1$ ball of fixed radius $r$ about $p$. Is there a good upper bound for the probability that $A\cap B$ is nonempty, in terms of $n$, $\ell$, and $r$? I am most interested in limiting behavior as $n\to\infty$ (in which case, obviously, $\ell$ and $r$ would have to depend on $n$).

Let $A$ be the subset of $\mathbb{R}^n$ defined by $A=\{x\in\mathbb{R}^{n}:|x_{1}-x_{n}|+\sum_{i=1}^{n-1}|x_{i+1}-x_{i}|\leq d\}$ for a given $d$. Next, sample a point $p$ uniformly in the unit cube, and let $B$ be the $\ell_1$ ball of fixed radius $r$ about $p$. Is there a good upper bound for the probability that $A\cap B$ is nonempty, in terms of $n$, $d$, and $r$? I am most interested in limiting behavior as $n\to\infty$ (in which case, obviously, $d$ and $r$ would have to depend on $n$).