# Variations on the Sum-Product Problem II

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

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

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#### 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 }$.