Partitions of positive integers into sets without infinite progressions

We prove a result which implies that, for any real numbers a and b satisfying 0 ≤ a ≤ b ≤ 1, there exists an infinite sequence of positive integers A with lower density a and upper density b such that the sets A and N \ A contain no infinite arithmetic and geometric progressions. Furthermore, for any m ≥ 2 and any positive numbers a1, . . . , am satisfying a1 + · · · + am = 1, we give an explicit partition of N into m disjoint sets ∪j=1Aj such that dP (Aj) = aj for each j = 1, . . . ,m and each infinite arithmetic and geometric progression P, where dP (Aj) denotes the proportion between the elements of P that belong to Aj and all elements of P, if a corresponding limit exists. In particular, for a = 1/2 and m = 2, this gives an explicit partition of N into two disjoint sets such that half of elements in each infinite arithmetic and geometric progression will be in one set and half in another.