Improved bounds on the set A(A+1)

For a subset A of a field F, write A(A+1) for the set {a(b+1): a,b∈A}. We establish new estimates on the size of A(A+1) in the case where F is either a finite field of prime order, or the real line.In the finite field case we show that A(A+1) is of cardinality at least |A|57/56−o(1), so long as |A|