Bugeaud Collection of Conjectures and Open Questions: Fractional Parts of Powers

Waldschmidt [Wal03] conjectured that a stronger result holds, namely that there exists a positive real number~$c$ such that $\lVert e^n \rVert > n^{-c}$ for every~$n \ge 2$. This is supported by metrical results [Kok45].

Note: the bound $n^{-c}$ equals $1$ when $n = 1$ for all $c$, while the distance to the nearest integer is always at most $1/2$, so the conjecture must start at $n \ge 2$.