Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A quantified version of Bourgain’s sum-product estimate in Fp for subsets of incomparable sizes. | A quantified version of Bourgain s sum-product estimate in Fp for subsets of incomparable sizes M. Z. Garaev Institute de Matemáticas Universidad Nacional Autánoma de Máxico Campus Morelia Apartado Postal 61-3 Xangari . 58089 Morelia Michoacán Máxico garaev@ Submitted Mar 4 2008 Accepted Apr 6 2008 Published Apr 18 2008 Mathematics Subject Classification 11B75 11T23 Abstract Let Fp be the field of residue classes modulo a prime number p. In this paper we prove that if A B c F then for any fixed 0 A A AB mm B A_ . 1 25_ A . This quantifies Bourgain s recent sum-product estimate. 1 Introduction Let Fp be the field of residue classes modulo a prime number p and let A be a non-empty subset of Fp. It is known from 4 5 that if A p1-5 where Ỗ 0 then one has the sum-product estimate A A AA A l 1 for some s ố 0. This estimate and its proof has been quantified and simplified in 3 6 11 . Improving upon our earlier estimate from 6 Katz and Shen 11 have shown that in the most nontrivial range 1 A p1 2 one has A A AA A 11 13 log A O 1 . A version of sum-product estimates with subsequent application to exponential sum bounds is given in 3 . In particular from 3 it follows that if 1 A p12 23 then A - A AA A 13 12 log A O 1 . THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R58 1 We also mention that in the case IAI p2 3 one has max A A AA g p1 2 A 1 2 which is optimal in general settings bound apart from the value of the implied constant for the details see 7 . Sum-product estimates in Fp for different subsets of incomparable sizes have been obtained by Bourgain 1 . More recently he has shown in 2 that if A B c Fp then A A AB imin B tAi0 A 2 A for some absolute positive constant c. In the present paper we prove the following explicit version of this result. Theorem 1. For any non-empty subsets A B c Fp and any 0 we have 1 25 A where the implied constant may depend only on . Remark. One can expect that appropriate adaptation of techniques of 3 and 11 may lead to .
