Good functions, measures, and the Kleinbock–Tomanov conjecture

In this paper, we prove a conjecture of Kleinbock and Tomanov (2007) on Diophantine properties of a large class of fractal measures on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="double-struck">Q</m:mi> <m:mi>p</m:mi> <m:mi>n</m:mi> </m:msubsup> </m:math> Qₚⁿ . More generally, we establish the 𝑝-adic analogues of the influential results of Kleinbock, Lindenstrauss and Weiss (2004) on Diophantine properties of friendly measures. We further prove the 𝑝-adic analogue of one of the main results of Kleinbock (2008) concerning Diophantine inheritance of affine subspaces, which answers a question of Kleinbock. One of the key ingredients in the proofs of Kleinbock, Lindenstrauss and Weiss is a result on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>C</m:mi> <m:mo>,</m:mo> <m:mi>α</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (C,α) -good functions whose proof crucially uses the Mean Value Theorem. Our main technical innovation is an alternative approach to establishing that certain functions are <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>C</m:mi> <m:mo>,</m:mo> <m:mi>α</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (C,α) -good in the 𝑝-adic setting. We believe this result will be of independent interest.