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on Waldschmidt's Conjecture on the Distance from $e^n$ to the Nearest Integer

The statement

For a real number $x$ , let $\lVert x \rVert$ denote the distance from $x$ to the nearest integer, defined as $\lVert x \rVert = \min_{m \in \mathbb{Z}} |x - m|$ . Prove that there exists a positive real number $c$ such that for every integer $n \geq 2$ , the inequality $\lVert e^n \rVert > n^{-c}$ holds.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.