The inverse theorem for the nonlinear Roth configuration: an exposition

加性组合学中的罗思型定理

Roth-type theorems in additive combinatroics

本文探讨加性组合学中罗思型定理的发展，介绍了其起源、核心问题及主要研究成果。

Quantitative bounds in the nonlinear Roth theorem

We show that there exists $c>0$ such that any subset of $\{1, \dots, N\}$ 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 Szemerédi 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…

非线性Roth定理的定量界

On the Quantitative Bounds of Nonlinear Roth Theorem

该研究针对不含非平凡多项式进展$(x, x + y, x + y^2)$的子集$A \subseteq \{1, \dots, N\}$，推导了其大小的定量上界，通过发展密度增量框架、多线性转移原理等技术完成证明，为非线性Roth定理提供了严格的定量界。

Yet another proof of Szemeredi's theorem

Using the density-increment strategy of Roth and Gowers, we derive Szemeredi's theorem on arithmetic progressions from the inverse conjectures GI(s) for the Gowers norms, recently established by the authors and Ziegler.