  We show injectivity of the X-ray transform and the $d$-plane Radon transform
for distributions on the $n$-torus, lowering the regularity assumption in the
recent work by Abouelaz and Rouvi\`ere. We also show solenoidal injectivity of
the X-ray transform on the $n$-torus for tensor fields of any order, allowing
the tensors to have distribution valued coefficients. These imply new
injectivity results for the periodic broken ray transform on cubes of any
dimension.
