We show that affine subspaces of Euclidean space are of Khintchine type for divergence under certain multiplicative Diophantine conditions on the parameterizing matrix. This provides evidence towards the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence, or that Khintchine’s theorem still holds when restricted to the subspace. This result is proved as a special case of a more general Hausdorff measure result from which the Hausdorff dimension of [Formula: see text] intersected with an appropriate subspace is also obtained.
We show that affine subspaces of Euclidean space are of Khintchine type for divergence under certain multiplicative Diophantine conditions on the parametrizing matrix. This provides evidence towards the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence, or that Khintchine's theorem still holds when restricted to the subspace. This result is proved as a special case of a more general Hausdorff measure result from which the Hausdorff dimension of W (τ ) intersected with an appropriate subspace is also obtained.
Finally, I must of course thank Josephine, the light of my life. I started working on this research before she was born, wrote the bulk of my paper while listening to the boops and bings of her monitors in the NICU, and wrote this thesis while she played on and around me. The past year has been a tragic whirlwind for all of humanity, but for me it has been a blessing to finish this work while being able to watch her grow and learn. I will treasure those memories for the rest of my life.
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