# Variations on the Sum-Product Problem II

Research paper by Brendan Murphy, Oliver Roche-Newton, Ilya D. Shkredov

Indexed on: 21 Dec '17Published on: 30 Aug '17Published in: SIAM Journal on Discrete Mathematics

#### Abstract

SIAM Journal on Discrete Mathematics, Volume 31, Issue 3, Page 1878-1894, January 2017. This paper is a sequel to a paper entitled Variations on the sum-product problem by the same authors [SIAM J. Discrete Math., 29 (2015), pp. 514-540]. In this sequel, we quantitatively improve several of the main results of the first paper as well as generalize a method from it to give a near-optimal bound for a new expander. The main new results are the following bounds, which hold for any finite set $A \subset \mathbb R$: $\exists a \in A$ such that $A(A+a) \gtrsim A ^{\frac{3}{2}+\frac{1}{186}}, A(A-A) \gtrsim A ^{\frac{3}{2}+\frac{1}{34}}, A(A+A) \gtrsim A ^{\frac{3}{2}+\frac{5}{242}}, \{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i \in A \} \gg \frac{ A ^2}{\log A }$.