jats:titleAbstract</jats:title>jats:pThe purpose of this paper is to show an unexpected connection between Diophantine approximation and the behavior of waves on black hole interiors with negative cosmological constant jats:inline-formulajats:alternativesjats:tex-math$$\Lambda <0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miΛ</mml:mi> mml:mo<</mml:mo> mml:mn0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and explore the consequences of this for the Strong Cosmic Censorship conjecture in general relativity. We study linear scalar perturbations jats:inline-formulajats:alternativesjats:tex-math$$\psi $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miψ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> of Kerr–AdS solving jats:inline-formulajats:alternativesjats:tex-math$$\Box _g\psi -\frac{2}{3}\Lambda \psi =0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:msub mml:mo□</mml:mo> mml:mig</mml:mi> </mml:msub> mml:miψ</mml:mi> mml:mo-</mml:mo> mml:mfrac mml:mn2</mml:mn> mml:mn3</mml:mn> </mml:mfrac> mml:miΛ</mml:mi> mml:miψ</mml:mi> mml:mo=</mml:mo> mml:mn0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> with reflecting boundary conditions imposed at infinity. Understanding the behavior of jats:inline-formulajats:alternativesjats:tex-math$$\psi $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miψ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> at the Cauchy horizon corresponds to a linear analog of the problem of Strong Cosmic Censorship. Our main result shows that if the dimensionless black hole parameters mass jats:inline-formulajats:alternativesjats:tex-math$${\mathfrak {m}} = M \sqrt{-\Lambda }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mim</mml:mi> mml:mo=</mml:mo> mml:miM</mml:mi> mml:msqrt mml:mrow mml:mo-</mml:mo> mml:miΛ</mml:mi> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and angular momentum jats:inline-formulajats:alternativesjats:tex-math$${\mathfrak {a}} = a \sqrt{-\Lambda }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mia</mml:mi> mml:mo=</mml:mo> mml:mia</mml:mi> mml:msqrt mml:mrow mml:mo-</mml:mo> mml:miΛ</mml:mi> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> satisfy a certain non-Diophantine condition, then perturbations jats:inline-formulajats:alternativesjats:tex-math$$\psi $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miψ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> arising from generic smooth initial data blow up jats:inline-formulajats:alternativesjats:tex-math$$|\psi |\rightarrow +\infty $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mo|</mml:mo> mml:miψ</mml:mi> mml:mo|</mml:mo> mml:mo→</mml:mo> mml:mo+</mml:mo> mml:mi∞</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> at the Cauchy horizon. The proof crucially relies on a novel resonance phenomenon between stable trapping on the black hole exterior and the poles of the interior scattering operator that gives rise to a small divisors problem. Our result is in stark contrast to the result on Reissner–Nordström–AdS (Kehle in Commun Math Phys 376(1):145–200, 2020) as well as to previous work on the analogous problem for jats:inline-formulajats:alternativesjats:tex-math$$\Lambda \ge 0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miΛ</mml:mi> mml:mo≥</mml:mo> mml:mn0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>—in both cases such linear scalar perturbations were shown to remain bounded. As a result of the non-Diophantine condition, the set of parameters jats:inline-formulajats:alternativesjats:tex-math$${\mathfrak {m}}, {\mathfrak {a}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mim</mml:mi> mml:mo,</mml:mo> mml:mia</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> for which we show blow-up forms a Baire-generic but Lebesgue-exceptional subset of all parameters below the Hawking–Reall bound. On the other hand, we conjecture that for a set of parameters jats:inline-formulajats:alternativesjats:tex-math$${\mathfrak {m}}, {\mathfrak {a}} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mim</mml:mi> mml:mo,</mml:mo> mml:mia</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> which is Baire-exceptional but Lebesgue-generic, all linear scalar perturbations remain bounded at the Cauchy horizon jats:inline-formulajats:alternativesjats:tex-math$$|\psi |\le C$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mo|</mml:mo> mml:miψ</mml:mi> mml:mo|</mml:mo> mml:mo≤</mml:mo> mml:miC</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. This suggests that the validity of the jats:inline-formulajats:alternativesjats:tex-math$$C^0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:miC</mml:mi> mml:mn0</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>-formulation of Strong Cosmic Censorship for jats:inline-formulajats:alternativesjats:tex-math$$\Lambda <0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miΛ</mml:mi> mml:mo<</mml:mo> mml:mn0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> may change in a spectacular way according to the notion of genericity imposed.</jats:p>