Geodesic
On product sets of arithmetic progressions
Discrete analysis (2023)

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arXiv
We prove that the size of the product set of any finite arithmetic progression $\mathcal{A}\subset \mathbb{Z}$ satisfies \[|\mathcal A \cdot \mathcal A| \ge \frac{|\mathcal A|^2}{(\log |\mathcal A|)^{2θ+o(1)} } ,\] where $2θ=1-(1+\log\log 2)/(\log 2)$ is the constant appearing in the celebrated Erdős multiplication table problem. This confirms a conjecture of Elekes and Ruzsa from about two decades ago. If instead $\mathcal{A}$ is relaxed to be a subset of a finite arithmetic progression in integers with positive constant density, we prove that \[|\mathcal A \cdot \mathcal A | \ge \frac{|\mathcal A|^{2}}{(\log |\mathcal A|)^{2\log 2- 1 + o(1)}}. \] This solves the typical case of another conjecture of Elekes and Ruzsa on the size of the product set of a set $\mathcal{A}$ whose sumset is of size $O(|\mathcal{A}|)$. Our bounds are sharp up to the $o(1)$ term in the exponents. We further prove asymmetric extensions of the above results.
Publié le : 
Max Wenqiang Xu; Yunkun Zhou. On product sets of arithmetic progressions. Discrete analysis (2023). http://geodesic.mathdoc.fr/item/DAS_2023_a12/
@article{DAS_2023_a12,
     author = {Max Wenqiang Xu and Yunkun Zhou},
     title = {On product sets of arithmetic progressions},
     journal = {Discrete analysis},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2023_a12/}
}
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