Large subsets of 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 (Zmn,+). Let rk(Zmn) denote the maximal size of a subset of Zmn 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 rk(Zmn):If k≥ 5 is odd and P^-(m) ≥ (k+ 2) / 2 , then (Formula presented.)If k≥ 4 is even, P^-(m) ≥ k and m≡-1modk, then (Formula presented.) Moreover, we give some further improved lower bounds on rk(Zpn) for primes p≤ 31 and progression lengths 4 ≤ k≤ 8.