2016/11/23 by Shanshan Du, Hao Pan, Du, Shanshan +1
Computer Science · Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1611.07792
openalex publication_date 2016/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let q be an odd prime power. Combining the discussion of Varnavides and a recent theorem of Ellenberg and Gijswijt, we show that a subset A⊂\mathbb Fqn will contain many non-trivial three-term arithmetic progressions, whenever |A|≥ (cq q)n for some constant cq>0. After the first version of our manuscript was uploaded in the arXiv, we learned from Professors Jacob Fox and Terence Tao that our result is a special case of a result of Fox and Lovasz [1, Theorem 3]. In fact, [1, Theorem 3] gives a much better bound than ours. For example, when q=3, the lower bound given by Fox and Lovasz is |A|2⋅ (|A|q-n)11.901, while our bound is |A|2⋅ (|A|q-n)25.803. We thank Professors Jacob Fox and Terence Tao for their helpful comments on our manuscript. [1] Jacob Fox, László Miklós Lovász, A tight bound for Green's arithmetic triangle removal lemma in vector spaces, preprint, arXiv:1606.01230.