A polylogarithmic bound in the nonlinear Roth theorem

The inverse theorem for the nonlinear Roth configuration: an exposition

We give an exposition of the inverse theorem for the cut-norm associated to the nonlinear Roth configuration, established previously by Peluse and the author.

在无 $K_{s,t}$ 子集中的 Roth 型定理

Roth-type theorems in $K_{s,t}$-free sets

该研究证明了在无 $K_{s,t}$ 子集中的 Roth 型定理，并在有限域向量空间中获得了更强的定量界。

一个热门多维非线性Roth定理中的界

Bounds in a popular multidimensional nonlinear Roth theorem

本研究得到非线性Roth定理的多维版本，为相关多维多项式Szemerédi定理情形的有效化迈出第一步，还证明了其有效热门版本，且该结果的量化依赖为指数型，异于线性情形的塔型界。

Nonlinear Roth type theorems in finite fields

We obtain smoothing estimates for certain nonlinear convolution operators on prime fields, leading to quantitative nonlinear Roth type theorems. Compared with the usual linear setting (i.e. arithmetic progressions), the nonlinear nature of the operators leads to different phenomena, both qualitatively and quantitatively.

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。