New bounds for dimensions of a set uniformly avoiding multi-dimensional arithmetic progressions

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.