A note on the set A(A + A)

Let p be a large enough prime number. When A is a subset of Fp\{0} of cardinality |A|>(p+1)∕3, then an application of the Cauchy–Davenport theorem gives Fp\{0}⊂A(A+A). In this note, we improve on this and we show that |A|≥0.3051p implies A(A+A)⊇Fp\{0}. In the opposite direction we show that there exists a set A such that |A|>(18+o(1))p and Fp\{0}⊈A(A+A).