Large Subsets of ${\mathbb{Z}}_m^n$ without Arithmetic Progressions

For integers $m$ and $n$, we study the problem of finding good lower bounds for the size of progression-free sets in $({\mathbb{Z}}_m^n,+)$. Let $r_k({\mathbb{Z}}_m^n)$ denote the maximal size of a subset of ${\mathbb{Z}}_m^n$ without arithmetic progressions of length $k$ and let $P^{-}(m)$ denote the least prime factor of $m$. We construct explicit progression-free sets and obtain the following improved lower bounds for $r_k({\mathbb{Z}}_m^n)$: If $k\ge 5$ is odd and $P^{-}(m)\ge (k+2)\mathrm{/}2$, then $$r_k({\mathbb{Z}}_m^n){\gg }_{m,k}\frac{\lfloor \frac{k-1}{k+1}m+1{\rfloor }^n}{n^{\lfloor \frac{k-1}{k+1}m\rfloor \mathrm{/}2}}\mathrm{.}$$ If $k\ge 4$ is even, $P^{-}(m)\ge k$ and $m\equiv -1k$, then $$r_k({\mathbb{Z}}_m^n){\gg }_{m,k}\frac{\lfloor \frac{k-2}{k}m+2{\rfloor }^n}{n^{\lfloor \frac{k-2}{k}m+1\rfloor \mathrm{/}2}}\mathrm{.}$$ Moreover, we give some further improved lower bounds on $r_k({\mathbb{Z}}_p^n)$ for primes $p\le 31$ and progression lengths $4\le k\le 8$.