  We study how small a local set of the continuum Gaussian free field (GFF) in
dimension $d$ has to be to ensure that this set is thin, which loosely speaking
means that it captures no GFF mass on itself, in other words, that the field
restricted to it is zero. We provide a criterion on the size of the local set
for this to happen, and on the other hand, we show that this criterion is sharp
by constructing small local sets that are not thin.
