An Explicit Sum-Product Estimate in p

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Growth in SL_3(Z/pZ) (2008, methodology): “, Cor. 6.29) suﬃce to derive from (3.6) the conclusion that |A· A+A· A| ≥ |A|· ( 1 2 min(|A|,p/|A|))1/6 for every subset Aof F∗ p. An application of the Katz-Tao lemma ([TV, Lem. 2.53]; see also [B], [Ga]) then suﬃces to show that, for every A⊂ F∗ p with |A| <p1−δ, δ>0, we have either |A+A| &g”
An improved sum-product inequality in fields of prime order (2010, background): “Garaev [3] succeeded in obtaining the first quantitative sum-product estimate in Fp: either the sum or the product set shall have cardinality |A|1+ 1 14 , up to a multiple of a power of c log |A|.”
An explicit incidence theorem in p (2010, background): “turn to Fp for the rest of the paper. On the level of the existence of positive exponents, a nontrivial sum-productestimate implies a non-trivial incidence or Becktype theorem, and conversely. Garaev [6] succeeded in obtaining a quantitative sumproduct estimate in Fp: for a small enough (say |A| &lt”
On the size of the set A(A + 1) (2008, background): “] that if |A| < p1−δ, where δ > 0, then one has the sum-product estimate |A+A|+|AA| ≫ |A|1+ε; ε = ε(δ) > 0. This estimate and its proof consequently have been quantiﬁed and simpliﬁed in [2], [6]–[9], [11]–[15], [17]. From the sum-product estimate and Ruzsa’s triangle inequalities (see, [13]”