An upper bound of the Hausdorff dimension of singular vectors on affine subspaces

In Diophantine approximation, the notion of singular vectors was introduced by Khintchine in the 1920’s. We study the set of singular vectors on an affine subspace of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We give an upper bound of its Hausdorff dimension in terms of the Diophantine exponent of the parameter of the affine subspace.