For a positive integer $\ n\ $ , define $\ x:=e^n\ $ and $\ y\ $ the number we get if we round $x$ correctly to the next integer.
Can $\ |x-y|\ $ get arbitrarily small ? In other words, are there better and better "near"-integers of the form $\ e^n\ $ with positive integers $\ n\ $ ?
The smallest value up to $\ n=10^4\ $ occurs for $\ n=5469\ $ with fractional part $\ 0.000029\cdots $
If we could show that the set {$frac(e^n)\mid n\in \mathbb N$} is dense in $\ [0,1]\ $ , this would answer my question. Can we do that ?
