  This article is concerned with the study of the fractal dimension of thick
points for a 4-dimensional Gaussian Free Field. We adopt the definition of
Gaussian Free Field on $\R^4$ introduced by Chen and Jakobson (2012) viewed as
an abstract Wiener space with underlying Hilbert space $H^2(\R^4)$. We can
prove that for $0\leq a \leq 4$ the Hausdorff dimension of the set of $a$-high
points is $4-a$. We also show that the set of thick points gives full mass to
the 4-dimensional Liouville Quantum Gravity measure.
