\lesssim \frac{N}{C^2}.$

By taking $C$ to be a large constant, an immediate corollary is:

Let $A \subset \C$ be a finite set. Then $$| A+ A\cdot A| = |\{ a + bc \,|\, a,b,c \in A\}| \gtrsim \frac{|A|^{1.5}}{\sqrt{\log |A|}}.$$

Organization
------------

In Section \[sec-prelim\] we give some preliminaries, including on design matrices and the Veronese embedding. In Section \[sec-manyrichlines\] we prove Theorem \[thm-manyrichlines\]. In Section \[sec-apbound\] we prove Theorem \[thm-apbound\]. In Section \[sec-newsumproduct\] we prove Theorem \[thm-newsumproduct\]. In Appendix \[sec-stronger-thm1\] we give a possible strengthening of Conjecture \[conj-rich\] along with the proofs of Theorem \[thm-conj-d=3\] and Theorem \[thm-conj-d=4\].

Acknowledgements
----------------

We thank Ben Green and Noam Solomon for helpful comments. Research supported by NSF grant CCF-1217416 and by the Sloan fellowship. Some of the work on the paper was carried out during the special semester on ‘Algebraic Techniques for Combinatorial and Computational Geometry’, held at the Institute for Pure and Applied Mathematics (IPAM) during Spring 2014.

Preliminaries {#sec-prelim}
=============

We begin with some notations. For a vector $v \in \C^n$ and a set $I \subset [n]$ we denote by $v_I \subset \C^I$ the restriction of $v$ to indices in $I$. We denote the [*support*]{} of a vector $v \in \C^d$ 