格雷厄姆重排猜想的Kneser化反集中与逆吸收
Kneserized Anticoncentration and Reverse Absorption for Graham's Rearrangement Conjecture
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中文总结 AI 辅助
该研究建立复合循环群的Kneser化反集中估计，结合已知结果证明格雷厄姆重排猜想对该群族成立，通过逆吸收方法推广到特定形式的循环群子集。
中文摘要 AI 辅助
我们为复合循环群中的均匀子集和建立了基于Kneser的反集中估计。该估计存在周期性损失，且弱于其素数模对应估计。不过，结合已知的小集与大集结果，它证明了：对每个满足ℤ_t是强可序列的固定t≥2及每个充分大素数p，ℤ_{tp}\{0}的每个子集都存在有效排序，从而为该复合循环群族建立了格雷厄姆重排猜想的类似结论。随后我们确定了该损失的结构来源：一个逆定理表明，素数型反集中所依赖的无稳定子增长的失效，会迫使集合的几乎所有元素落入某个真子群或其一个陪集。我们通过逆吸收（reverse absorption）来利用这一结构。将非周期反集中与结构化集中之间的二分法迭代，可证明当L与γ固定且素数p_i充分大时，每个满足k=∏_{i=1}^{s}p_i^{e_i}、∑_{i=1}^{s}e_i≤L、p_1<…<p_s≤γp_1的ℤ_k\{0}都存在有效排序。
英文摘要
We establish a Kneser-based anticoncentration estimate for uniform subset sums in composite cyclic groups. The estimate contains a periodic loss and is weaker than its prime-modulus counterpart. Nevertheless, together with known small- and large-set results, it proves that, for every fixed $t\geq2$ such that $\mathbb{Z}_t$ is strongly sequenceable and every sufficiently large prime $p$, every subset of $\mathbb{Z}_{tp}\setminus\{0\}$ has a valid ordering, thus establishing the analogue of Graham's rearrangement conjecture for this family of composite cyclic groups. We then identify the structural source of this loss. An inverse theorem shows that failure of the stabilizer-free growth underlying prime-type anticoncentration forces almost all of the set into a proper subgroup or one of its cosets. We exploit this structure by reverse absorption. Iterating the resulting dichotomy between non-periodic anticoncentration and structured concentration proves that every subset of \[ \mathbb{Z}_k\setminus\{0\}, \qquad k=\prod_{i=1}^{s}p_i^{e_i}, \qquad \sum_{i=1}^{s}e_i\leq L, \qquad p_1<\cdots<p_s\leqγp_1, \] admits a valid ordering whenever $L$ and $γ$ are fixed and the primes $p_i$ are sufficiently large.
