Quantitative bounds in the nonlinear Roth theorem

• 成果类型: Article
• 署名作者: Peluse, Sarah; Prendiville, Sean
• 署名单位: Stanford University; Lancaster University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01293-x
• 发表日期: 2024
• 页码: 865-903
• 关键词: difference sets; szemeredi theorem; integer sets; progressions; SEQUENCES; PROOF

摘要

We show that there exists c>0 such that any subset of {1, ... ,N} of density at least (log log N)(-c) contains a nontrivial progression of the form x, x+y, x+y(2). This is the first quantitatively effective version of the Bergelson-Leibman polynomial Szemeredi theorem for a progression involving polynomials of differing degrees. Our key innovation is an inverse theorem characterising sets for which the number of configurations x, x+y, x+y(2) deviates substantially from the expected value. In proving this, we develop the first effective instance of a concatenation theorem of Tao and Ziegler, with polynomial bounds.