  Let $r_k(N)$ be the largest cardinality of a subset of $\{1,\ldots,N\}$ which
does not contain any arithmetic progressions (APs) of length $k$. In this
paper, we give new upper and lower bounds for fractal dimensions of a set which
does not contain $(k,\epsilon)$-APs in terms of $r_k(N)$, where $N$ depends on
$\epsilon$. Here we say that a subset of real numbers does not contain
$(k,\epsilon)$-APs if we can not find any APs of length $k$ with gap difference
$\Delta$ in the $\epsilon \Delta$-neighborhood of the set. More precisely, we
show multi-dimensional cases of this result. As a corollary, we find
equivalences between multi-dimensional Szemer\'edi's theorem and bounds for
fractal dimensions of a set which does not contain multi-dimensional
$(k,\epsilon)$-APs.
