Result: ON THE NUMBER OF SUMS AND PRODUCTS: On the number of sums and products

For a finite set \(A\) of complex numbers, the sum-set \(A+A\) and the product-set \(A\cdot A\) are defined respectively by \(A+A = \{a+b: a,b \in A\}\) and \(A\cdot A= \{a\cdot b: a,b \in A\}\). Towards a conjecture of Erdős and Szemerédi, the following new lower bound has been established in the present paper: \(\text{ max} (| A+A| , | A\cdot A| ) \geq \frac{cn^{14/11}}{\log^{3/11} n}\), where \(c\) is a positive absolute constant. Earlier results in this direction were obtained by \textit{P. Erdős} and \textit{E. Szemerédi} [Studies in pure mathematics, Mem. of P. Turan, 213--218 (1983; Zbl 0526.10011)], \textit{M. B. Nathanson} [Proc. Am. Math. Soc. 125, 9--16 (1997; Zbl 0869.11010)], \textit{K. Ford} [Ramanujan J. 2, 59-66 (1998; Zbl 0908.11008)] and \textit{G. Elekes} [Acta Arith. 81, 365--367 (1997; Zbl 0887.11012)]. A corollary of the main result gives a lower bound of the size of the product-set, if that of the sum-set is at most \(c| A| \). Earlier results in this direction were given by \textit{M. B. Nathanson} and \textit{G. Tenenbaum} [Astérisque 258, 195--204 (1999; Zbl 0947.11008)], \textit{G. Elekes} and \textit{I. Z. Ruzsa} [Stud. Sci. Math. Hungar. 40, 301--308 (2003; Zbl 1102.11009)] and \textit{M. Chang} [Geom. Funct. Anal. 13, 720--736 (2003; Zbl 1029.11006)].