AI 中文总结
该研究针对Roth整数着色中点问题提出非对称变体，采用傅里叶分析方法，结合相关定理在有限域与整数环境中推导算术级数的渐近紧界并改进颜色数量的依赖关系。
AI 中文摘要
我们引入了Roth关于整数着色与中点问题的经典问题的非对称变体。该新问题无法使用Erdös-Sárközy-Sós的标准对称方法解决。我们描述了一种傅里叶分析方法，该方法给出了渐近紧界。在$\boldsymbol{\text{F}}_3^n$环境中，我们证明的密度增量是有效的，结合Freiman-Ruzsa定理，可在区分集中提供一个近最优余维数的仿射子空间。该方法通过Bohr集和Bogolyubov-Ruzsa定理应用于从1到N的整数环境，得到一个长度为N的幂的级数，其仅依赖于颜色数量。随后，我们使用Chang引理和Balog-Szemerédi-Gowers定理进一步改进了对颜色数量的依赖关系。
英文摘要
We introduce an asymmetric variant of a classic problem by Roth about colourings of integers and midpoints. The new problem cannot be solved with the standard approach by Erdös-Sárközy-Sós which uses symmetry. We describe a Fourier Analytic approach which gives asymptotically tight bounds. In the $\mathbb{F}_3^n$ setting, the density increment we show is efficient and together with the Freiman-Ruzsa Theorem provides an affine subspace of near optimal codimension within the distinguished set. This approach adopted to the setting of the integers from $1$ to $N$ via Bohr sets and Bogolyubov-Ruzsa Lemma gives a progression of length being a power of $N$ which only depends on the number of colours. We then use Chang's Lemma and Balog-Szemerédi-Gowers Theorem to further improve the dependence on the number of colours.
