AbstractDiophantine approximation is fundamentally concerned with determining the size ofsets of irrational points which can be approximated by rational ones to a high enoughdegree. One classical theorem, Khintchine’s Theorem, gives a zero one law for the set ofnumbers which can be approximated “ψ-well”. This theorem can be extended to higherdimensions, and there it makes sense to ask if the typical point on a measure zerosubspace behaves like a typical point in Euclidean space. A priori Khintchine’s theoremsays nothing about this, but strides have been made to determine which subspacesinherit Diophantine properties of the ambient space such asψ-approximability. Herewe show that affine subspaces of Euclidean space are of Khintchine type for divergenceunder certain multiplicative Diophantine conditions on the parametrizing matrix, orcertain Diophantine conditions on the shift vector in the event that the parametrizingmatrix is rational. This provides evidence towards the conjecture that all affine subspacesof Euclidean space are of Khintchine type for divergence, or that Khintchine’s theoremstill holds when restricted to the subspace. The first of these results is proved as a specialcase of a more general Hausdorfff-measure result from which the Hausdorff dimensionofW(τ) intersected with an appropriate subspace is also obtained