A note on the set \(A(A + A)\) (Q2420497)

Let \(F_p\) denote the field with \(p\) elements. The Cauchy-Davenport theorem implies that for a sufficiently large prime number \(p\), for all sufficiently large subsets \(A\subseteq F_p\setminus \{0\}\), namely for \(|A|> (p+1)/3\), we have \(F_p\setminus \{0\}\subseteq A(A+A) \). The paper under review studies how large \(|A|\) has to be to imply the conclusion above. The paper shows that \(|A|>0.3051p\) suffices, while some \(|A|> \Bigl(\frac{1}{8} +o(1)\Bigl)p\) does not.