More sum-product type counterexamples: products with shifts and $AA+A$

Abstract

Adapting the construction disproving the sum-product conjecture over $\mathbb{R}$ present in Bloom, Sawin, Schildkraut and Zhelezov, we show the existence of a constant $c>0$ and arbitrarily large finite sets $A\subseteq \mathbb{R}$ such that $$|AA+A+A|\ll |A{|}^{2-c}.$$ As a corollary, all of the sets $A+A$, $AA$, $(A+1)(A+1)$, $A(A+1)$ and $AA+A$ are of size $O(|A{|}^{2-c})$ for this construction.

Added Carl Schildkraut as an author, and added further expander results