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arXiv · 2606.24583

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

Oliver Roche-Newton·Carl Schildkraut·Audie Warren
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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.

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Oliver Roche-Newton, Carl Schildkraut, Audie Warren. 2026-06-23. More sum-product type counterexamples: products with shifts and $AA+A$. https://arxiv.org/abs/2606.24583

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