We prove that there exist infinitely many coprime numbers $a$, $b$, $c$ with $a+b=c$ and $c>\operatorname{rad}(abc)\exp(6.563\sqrt{\log c}/\log\log c)$. These are the most extremal examples currently known in the $abc$ conjecture, thereby providing a new lower bound on the tightest possible form of the conjecture. This builds on work of van Frankenhuysen (1999) who proved the existence of examples satisfying the above bound with the constant $6.068$ in place of $6.563$. We show that the constant $6.563$ may be replaced by $4\sqrt{2\delta/e}$ where $\delta$ is a constant such that all full-rank unimodular lattices of sufficiently large dimension $n$ contain a nonzero vector with $\ell_1$ norm at most $n/\delta$.

翻译：我们证明存在无穷多互素的数$a$、$b$、$c$满足$a+b=c$且$c>\operatorname{rad}(abc)\exp(6.563\sqrt{\log c}/\log\log c)$。这些是目前$abc$猜想中已知的最极端例子，从而为该猜想最精确形式提供了新的下界。该结果建立在van Frankenhuysen（1999）工作的基础之上，他证明了存在满足上述不等式的例子，其中常数$6.068$代替了$6.563$。我们表明常数$6.563$可以替换为$4\sqrt{2\delta/e}$，其中$\delta$是一个常数，使得所有充分大维数$n$的满秩幺模格都包含一个非零向量，其$\ell_1$范数不超过$n/\delta$。