On the reducibility of large sets of residues modulo p

It is shown that if p>2p>2 and CC is a subset of FpFp with |C|⩾p−C1plogp then there are A∈FpA∈Fp, B∈FpB∈Fp with C=A+BC=A+B, |A|⩾2|A|⩾2, |B|⩾2|B|⩾2. On the other hand, for every prime p   there is a subset C⊂FpC⊂Fp with |C|>p−C2loglogp(logp)1/2p such that there are no AA, BB with these properties.