A Sharp Entropy Condition for the Density of Angular Derivatives

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> be a holomorphic self-map of the unit disc. We show that if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">log</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mo>|</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> <mml:mo>|</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is integrable on a sub-arc of the unit circle, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>I</mml:mi> </mml:math> , then the set of points where the function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> has finite Carathéodory angular derivative on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>I</mml:mi> </mml:math> is a countable union of Beurling–Carleson sets of finite entropy. Conversely, given a countable union of Beurling–Carleson sets, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> , we construct a holomorphic self-map of the unit disc, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> , such that the set of points where the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> function has finite Carathéodory angular derivative is equal to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">log</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mo>|</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> <mml:mo>|</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is integrable on the unit circle. Our main technical tools are the Aleksandrov disintegration Theorem and a characterization of countable unions of Beurling–Carleson sets due to Makarov and Nikolski.