Large sets in finite fields are sumsets (Q996275)

For a prime number \(p>2\), let \(\mathbb Z_p\) denote the abelian group with \(p\) elements. A subset \(S\) of \(\mathbb Z_p\) is called a sumset if there is a set \(A\subset\mathbb Z_p\) so that \(A+A=\{a_1+a_2: a_1,a_2 \in A\}=S\). Let \(f(p)\) denote the maximum integer \(f\) so that every \(S\subset\mathbb Z_p\) of size at least \(p-f\) is a sumset. The author proves that there exist two positive constants \(c_1, c_2\) and an integer \(p_0\) so that for all \(p>p_0\) \[ c_1\frac{ \sqrt{p}}{\sqrt{\log p}}\leq f(p)<c_2\frac{p^{\frac23}}{(\log p)^{\frac13}}. \]