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 ( Z m n , + ) . Let r k ( Z m n ) denote the maximal size of a subset of 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 ( Z m n ) :If k ≥ 5 is odd and P - ( m ) ≥ ( k + 2 ) / 2 , then r k ( Z m n ) ≫ m , k ⌊ k - 1 k + 1 m + 1 ⌋ n n ⌊ k - 1 k + 1 m ⌋ / 2 . If k ≥ 4 is even, P - ( m ) ≥ k and m ≡ - 1 mod k , then r k ( Z m n ) ≫ m , k ⌊ k - 2 k m + 2 ⌋ n n ⌊ k - 2 k m + 1 ⌋ / 2 . Moreover, we give some further improved lower bounds on r k ( Z p n ) for primes p ≤ 31 and progression lengths 4 ≤ k ≤ 8 .