One construction of a K3 surface with a dense set of rational points
  We prove that there exists a number field $\fie$ and a smooth projective $\mathrm{K3}$ surface $S_{22}$ (of genus $12$) over $\fie$ such that the geometric Picard number of $S_{22}$ is equal to $1$ and the $\fie$-rational points of $S_{22}$ are Zariski dense. 
Ilya Karzhemanov
math.AG math.NT