On a Lower Bound for $\|(4/3)^k\|$
  We prove, that $$ \biggl\|\biggl(\frac43\biggr)^k\biggr\| >\biggl(\frac49\biggr)^k\qquad\text{for}\quad k\ge6, $$ where $\|\cdot\|$ is a distance to the nearest prime. 
Yury Pupyrev
math.NT