2019/12/07 by Lev, Vsevolod F., Shkredov, Ilya D.
#11P70 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1912.03483
Improving upon the results of Freiman and Candela-Serra-Spiegel, we show that for a non-empty subset A⊆\mathbb Fp with p prime and |A|<0.0045p, (i) if |A+A|<2.59|A|-3 and |A|>100, then A is contained in an arithmetic progression of size |A+A|-|A|+1, and (ii) if |A-A|<2.6|A|-3, then A is contained in an arithmetic progression of size |A-A|-|A|+1. The improvement comes from using the properties of higher energies.