A finite set $S \subset \mathbb{Z}$ is a Sidon set if its pairwise differences are distinct. Recall that a perfect difference set (PDS) of order $n$ is a set $B \subset \mathbb{Z}_v$ ($v = n^2 - n + 1$) of size $n$ such that every nonzero residue arises exactly once as a difference of two elements of $B$. Erdős's \$1000 conjecture -- that every finite Sidon set extends to a finite PDS -- was disproved by Alexeev and Mixon (arXiv:2510.19804, October 2025), via the size-5 counterexamples $\{1,2,4,8,13\}$ and Hall's earlier $\{1,3,9,10,13\}$; they then asked: what is the smallest size $s$ of a non-extending Sidon set? The trivial bounds give $3 \le s \le 5$. Our evidence points to $s = 4$. We exhibit two integer Sidon sets, \[ A = \{0, 1, 3, 11\}, \qquad B = \{0, 1, 4, 11\}, \] together with the apparent infinite family of dilations $kA$, $kB$ and their reflections, all of which fail to extend for every prime power $q \le 317$ via the Singer affine-orbit check (rigorous under Hall's 1947 uniqueness for Desarguesian cyclic planes through $q \le 40$ and under the prime-power conjecture beyond that), and unconditionally for every modulus $v \le 133$ via brute-force depth-first search. We also report the exact density $N_{\text{ne}}(N) = 4 \lfloor N / 11 \rfloor$ of non-extending size-4 Sidon sets in $[0, N]$ for $N \le 50$ -- the match is exact, which suggests the $kA, kB$ family is complete in this range. A complete proof, perhaps in the spirit of Alexeev--Mixon's polarity argument or via a multiplier descent, remains open.

翻译：有限集合$S \subset \mathbb{Z}$被称为Sidon集，如果其两两差互不相同。回顾一下，阶为$n$的完美差集（PDS）是指一个大小为$n$的集合$B \subset \mathbb{Z}_v$（其中$v = n^2 - n + 1$），使得每个非零剩余恰好作为$B$中两个元素的差出现一次。Erdős的1000美元猜想——每个有限Sidon集都能扩展为一个有限完美差集——已被Alexeev和Mixon（arXiv:2510.19804，2025年10月）通过大小为5的反例$\{1,2,4,8,13\}$以及Hall早先的$\{1,3,9,10,13\}$所否定；他们随后提出问题：不可扩展Sidon集的最小大小$s$是多少？平凡界限给出$3 \le s \le 5$。我们的证据指向$s = 4$。我们展示了两个整数Sidon集：\[ A = \{0, 1, 3, 11\}, \qquad B = \{0, 1, 4, 11\}, \]以及明显的无穷族膨胀$kA$、$kB$及其反射，所有这些集合在通过Singer仿射轨道检查（在$q \le 40$时基于Hall 1947年的Desarguesian循环平面唯一性严格成立，在此之上基于素数幂猜想）的每个素数幂$q \le 317$下均未能扩展，并且通过暴力深度优先搜索无条件地对每个模数$v \le 133$也未能扩展。我们还报告了在$N \le 50$范围内$[0, N]$中不可扩展的大小为4的Sidon集的确切密度$N_{\text{ne}}(N) = 4 \lfloor N / 11 \rfloor$——这一匹配是精确的，表明在此范围内$kA, kB$族是完备的。一个完整的证明，或许遵循Alexeev–Mixon的极性论证思路或通过乘数下降法，仍有待进一步研究。