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prove that for sets $A, B, C \\subset \\mathbb{F}_p$ with $|A|=|B|=|C| \\leq \\sqrt{p}$ and a fixed $0 \\neq d \\in \\mathbb{F}_p$ holds\n  $$\n  \\max(|AB|, |(A+d)C|) \\gg|A|^{1+1/26}.\n  $$\n  In particular,\n  $$\n  |A(A+1)| \\gg |A|^{1 + 1/26}\n  $$\n  and\n  $$\n  \\max(|AA|, |(A+1)(A+1)|) \\gg |A|^{1 + 1/26}.\n  $$\n  The first estimate improves the bound by Roche-Newton and Jones.\n  In the general case of a field of order $q = p^m$ we obtain similar estimates with the exponent $1+1/559 + o(1)$ under the condition that $AB$ does not have large intersection with any subfield coset, answering a question of Shp","authors_text":"Dmitrii Zhelezov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-20T16:07:15Z","title":"On additive shifts of multiplicative almost-subgroups in finite 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