# Variations on the sum-product problem II

## TL;DR

This paper improves bounds on sum-product type problems for real sets, establishing new lower bounds for certain product and sum sets, and analyzing complex polynomial-logarithm expressions, advancing the understanding of additive and multiplicative structures.

## Contribution

It provides quantitative improvements on sum-product estimates and introduces new bounds for related combinatorial expressions, building on previous methods.

## Key findings

- Existence of an element a in A with |A(A+a)| erences |A|^{3/2 + 1/186}
- Improved bounds for |A(A+A)| and |A(A-A)|
- Lower bound for (a_1+a_2+a_3+a_4)^2 + \u03bb a_5

## Abstract

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the methods therein. The main new results is that, for any finite set $A \subset \mathbb R$, there exists $a \in A$ such that $|A(A+a)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{186}}$. We give improved bounds for the cardinalities of $A(A+A)$ and $A(A-A)$. Also, we prove that $|\{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i \in A \}| \gg \frac{|A|^2}{\log |A|}$. The latter result is optimal up to the logarithmic factor.