2022/01/01 by Max Wenqiang Xu, Xu, Max Wenqiang, Yunkun Zhou +1
Mathematics · #Advanced Topology and Set Theory #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2201.00104
openalex publication_date 2022/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that the size of the product set of any finite arithmetic progression A⊂ ℤ satisfies |\mathcal A ⋅ \mathcal A| ≥ \frac|\mathcal A|2(log |\mathcal A|)2θ+o(1) , where 2θ=1-(1+loglog 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 A is relaxed to be a subset of a finite arithmetic progression in integers with positive constant density, we prove that |\mathcal A ⋅ \mathcal A | ≥ \frac|\mathcal A|2(log |\mathcal A|)2log 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 A whose sumset is of size O(|A|). Our bounds are sharp up to the o(1) term in the exponents. We further prove asymmetric extensions of the above results.