On exponential sums over multiplicative subgroups of medium size

In the paper we obtain some new upper bounds for exponential sums over multiplicative subgroups Γ⊆Fp⁎ having sizes in the range [pc1,pc2][pc1,pc2], where c1c1, c2c2 are some absolute constants close to 1/2. As an application we prove that in symmetric case Γ   is always an additive basis of order five, provided by |Γ|≫p1/2log1/3⁡p|Γ|≫p1/2log1/3⁡p. Also the method allows us to give a new upper bound for Heilbronn's exponential sum.