A note on expansion in prime fields

Original · EN

Let β,ε∈ (0,1], and k ≥ (122 {1/β,1/ε}). We prove that if A,B are subsets of a prime field Zₚ, and |B| ≥ pβ, then there exists a sum of the form S = a₁B ± ± aₖB, a₁,,aₖ ∈ A, with |S| ≥ 2⁻¹²p⁻ε{|A||B|,p}. As a corollary, we obtain an elementary proof of the following sum-product estimate. For every α< 1 and β,δ> 0, there exists ε> 0 such that the following holds. If A,B,E ⊂ Zₚ satisfy |A| ≤ pα, |B| ≥ pβ, and |B||E| ≥ pδ|A|, then there exists t ∈ E such that |A + tB| ≥ c pε|A|, for some absolute constant c > 0. A sharper estimate, based on the polynomial method, follows from recent work of Stevens and de Zeeuw.