2017/03/28 by Brendan Murphy, Murphy, Brendan, Oliver Roche‐Newton +5
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Number Theory (math.NT) #math.CO #math.NT
paper · pdf · doi:10.48550/arxiv.1703.09549
This paper supersedes arXiv:1603.06827
openalex publication_date 2017/03/28 · arxiv created 2017/04/04 · arxiv updated 2017/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the methods therein. The main new results is that, for any finite set A ⊂ \mathbb R, there exists a ∈ A such that |A(A+a)| \gtrsim |A|(3)/(2)+(1)/(186). We give improved bounds for the cardinalities of A(A+A) and A(A-A). Also, we prove that |\(a1+a2+a3+a4)2+log a5 : ai ∈ A \| ≫ (|A|2)/(log |A|). The latter result is optimal up to the logarithmic factor.