Long arithmetic progressions in A+A+AA+A+A with A a prime subset

If A   is a dense subset of the integers, then A+A+AA+A+A contains long arithmetic progressions. This problem has been studied by many people, but results of sparse sets are hard to obtain. In this paper, we prove that if A is a subset of the primes less than n   with cardinality αn/logn, then A+A+AA+A+A contains a long arithmetic progression of length at least{nc1α2/logα−1if α⩾(1/logloglogn)c0,c2α5(logα−1)−1nc3α4/logα−1if α⩾(1/logn)c0′.