Sets of integers that do not contain long arithmetic progressions

Combining ideas of Rankin, Elkin, Green & Wolf, we give constructive lower bounds for r(k)(N), the largest size of a subset of {1,2, ..., N} that does not contain a k-element arithmetic progression: For every epsilon > 0, if N is sufficiently large, then r(3)(N) >= N(6.2(3/4)root 5/e pi(3/2) - epsilon) exp(-root 8 log N + 1/4 log log N), r(k)(N) >= N C-k exp(-n2((n-1)/2) (n)root log N + 1/2n log log N), where C-k > 0 is an unspecified constant, log = log(2), exp(x) = 2(x), and n = [log VI. These are currently the best lower bounds for all k, and are an improvement over previous lower bounds for all k not equal 4.