Convexity and Sumsets

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New sum-product type estimates over finite fields (2014, methodology): “ving the sets 3A and 4A versus AA. These estimates can be compared with those for convex sets of reals, obtained by iteratively applying the Szemer´edi-Trotter theorem by Elekes, Nathanson and Ruzsa, [10]. We have: Corollary 12. Let A ⊂ Fq such that |A| < cp18/35. Then (16) |A +A +A|4|AA|9 ≫ |A|1”
NEW RESULTS ON SUM‐PRODUCT TYPE GROWTH OVER FIELDS (2017, methodology): “ then |A− A| ≫ |A|5/3−o(1). Both are instances of “breaking the 3/2 threshold.” In the former case the same estimate with |A|3/2 in the right-hand side was established by Elekes, Nathanson, and Ruzsa [11] from the Szemer´edi-Trotter theorem used in the form (2). In the latter case Heath-Brown and Ko”
On the size of k-fold sum and product sets of integers (2003, methodology): “ypeset by AMS-TEX 1 and in [So] a small further improvement |2A|+ |A(2)| >cN1411−ε. Again, using the Szemer´edi-Trotter theorem, Elekes-Nathanson-Ruzsa showed that |kA|·|A(k)| >c|A|3−21−k. (See [E-N-R].) The theorem proved in this paper answers aﬃrmatively a problem posed in their paper and pr”