无唯一和集的二阶对数下界
A Second-Logarithm Lower Bound for Sets with No Unique Sums
AI总结：
该论文针对奇素数p及有限阿贝尔群G，改进了无唯一和集的下界，证明了更强的二阶对数下界，相关结论已在Lean 4中验证，还得到了弱三元平衡集对应的上界。
AI中文摘要：
对于奇素数p，设m(p)为满足以下条件的集合A⊆Z/pZ的最小基数，其中|A|≥2，且A+A中没有和能以A中无序对（允许重复）的形式唯一表示。Bedert证明了m(p)≫log p·√log⁽³⁾p / log⁽⁴⁾p。我们证明了更强的下界m(p)≫log p·log log p。更一般地，若G是有限阿贝尔群，q(G)是|G|的最小素因子，则当q(G)>2时，同样的显式估计成立；特别地，当q(G)→∞时，每个满足|A|≥2且无唯一和的子集A⊆G的基数≫log q(G)·log log q(G)。证明有两个结构性输入：第一，A的一个最大子集，其大小不超过4的不同元素子集和均互不相同，该子集的基数≫log p，这由短坐标引理和碰撞格行列式论证得出；第二，我们改进了Bedert的密度增量，替代表示指向未覆盖端点，通过平移合并，并分为宽、暴露和重复三类，负载敏感熵引理利用重复平移的实际最终纤维重数对其编码，得到的全局平移集复杂度为exp(O(K))，其中K是|A|与四级加法维数的比值，这迫使K≫log log p，定理得证。所有主要结论及推导所用的结构性含义已在Lean 4中用显式整数常数验证。作为次要且逻辑独立的结果，我们构造了弱三元平衡集，并得到m(p)≤(log p)²/(2(log 3)²)+(2/log 3 + o(1))·(log p)²/log log p。
英文摘要：
For an odd prime $p$, let $m(p)$ be the minimum cardinality of a set $A\subseteq \mathbb Z/p\mathbb Z$, with $|A|\geq2$, such that no sum in $A+A$ has a unique representation as an unordered pair from $A$, with repetition allowed. Bedert proved \[ m(p)\gg \log p\, \frac{\sqrt{\log^{(3)}p}}{\log^{(4)}p}. \] We prove the stronger lower bound \[ m(p)\gg \log p\,\log\log p. \] More generally, if $G$ is a finite Abelian group and $q(G)$ is the least prime divisor of $|G|$, then the same explicit estimate holds whenever $q(G)>2$, and in particular every subset $A\subseteq G$ with $|A|\geq2$ and no unique sum has cardinality $\gg \log q(G)\,\log\log q(G)$ as $q(G)\to\infty$. The proof has two structural inputs. First, a maximum subset of $A$ whose distinct-element subset sums of size at most four are all different has cardinality $\gg\log p$. This follows from a short-coordinate lemma and a collision-lattice determinant argument. Second, we refine Bedert's density increment. Alternative representations are oriented toward an uncovered endpoint, coalesced by their translation, and separated into wide, exposed, and recurrent batches. A load-sensitive entropy lemma codes the recurrent translations using their actual final fibre multiplicities. The resulting global shift-set complexity is $\exp(O(K))$, where $K$ is the ratio of $|A|$ to the level-four additive dimension. This forces $K\gg\log\log p$, and the theorem follows. All headline statements and the structural implications used to derive them have also been checked in Lean~4 with explicit integer constants. As a secondary and logically independent result, we construct weakly ternary-balanced sets and obtain \[ m(p)\leq \frac{(\log p)^2}{2(\log 3)^2} +\left(\frac{2}{\log 3}+o(1)\right) \frac{(\log p)^2}{\log\log p}. \]
