$k$-配置的分支敏感密度增量

中文摘要 AI 辅助

我们细化了由Beker提出的用于$k$-配置的密度增量论证。重点不在于引入新的逆定理，而在于保留当图计数输出被压缩为单一均匀替代时丢失的信息。对于$k$-配置问题中出现的传递$K_k$，我们在图递归过程中保持一个固定矩，并获得三个真正不同的输出：行度异常、非中心网格异常和中心网格异常。中心情况通过相关Gram矩的正性直接处理，避免了通常的不平衡步骤。每个输出随后通过适当的局部Kelley–Meka机制进行路由，该机制具有自身的密度增益、秩成本和宽度成本。加权迭代预算防止最昂贵的局部分支被计费最大次数。对于奇数阶有限阿贝尔群$G$和密度为$\alpha$的$A\subseteq G$，记$L=\log(2/\alpha)$和$\Lambda=L+\log(2k)$。我们获得如下形式的下界：\\[ \Pp_{x_1,\ldots,x_k\in G}\\!\left(\frac{x_i+x_j}{2}\in A\\ \text{对于所有 }1\le i\le j\le k\right) \ge \exp\\!\bigl(-C\Phi(k,L,\Lambda)\bigr), \\] 其中 \\[ \Phi(k,L,\Lambda)=k^{36}L^6\Lambda^6+k^{34}L^8\Lambda^5+k^{32}L^{14}\Lambda+k^{30}L^{16}. \\] 因此，在Erdős–Moser应用中，可以将Beker的$k$-配置路线中的指数$68$替换为$36$；特别是，同样的缩减产生了每个指数$c<1/36$的无和集下界。

英文摘要

We refine the density-increment argument for $k$-configurations developed by Beker. The point is not to introduce a new inverse theorem, but to retain information that is lost when the graph-counting output is compressed into a single uniform alternative. For the transitive $K_k$ appearing in the $k$-configuration problem we keep a fixed moment throughout the graph recursion and obtain three genuinely different outputs: a row-degree anomaly, a non-centred grid anomaly, and a centred grid anomaly. The centred case is treated directly through positivity of the associated Gram moments, avoiding the usual unbalancing step. Each output is then routed through the appropriate local Kelley--Meka mechanism with its own density gain, rank cost, and width cost. A weighted iteration budget prevents the most expensive local branches from being charged the maximal number of times. For a finite abelian group $G$ of odd order and $A\subseteq G$ of density $α$, write $L=\log(2/α)$ and $Λ=L+\log(2k)$. We obtain a lower bound of the form \[ \Pp_{x_1,\ldots,x_k\in G}\!\left(\frac{x_i+x_j}{2}\in A\ \text{for all }1\le i\le j\le k\right) \ge \exp\!\bigl(-CΦ(k,L,Λ)\bigr), \] where \[ Φ(k,L,Λ)=k^{36}L^6Λ^6+k^{34}L^8Λ^5+k^{32}L^{14}Λ+k^{30}L^{16}. \] Consequently, in the Erdős--Moser application one may replace the exponent $68$ in Beker's $k$-configuration route by $36$; in particular the same reduction yields the sum-free lower bound with every exponent $c<1/36$.

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