Interpolating with generalized Assouad dimensions

The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϕ</mml:mi></mml:math> -Assouad dimensions are a family of dimensions which interpolate between the upper box and Assouad dimensions. They are a generalization of the well-studied Assouad spectrum with a more general form of scale sensitivity that is often closely related to "phase-transition" phenomena in sets. In this article we establish a number of key properties of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϕ</mml:mi></mml:math> -Assouad dimensions which help to clarify their behaviour. We prove for any bounded doubling metric space <i>F</i> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi></mml:mrow> </mml:math> satisfying <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub><mml:mover><mml:mtext>dim</mml:mtext> <mml:mo>¯</mml:mo></mml:mover> <mml:mtext>B</mml:mtext></mml:msub> <mml:mi>F</mml:mi> <mml:mo><</mml:mo> <mml:mi>α</mml:mi> <mml:mo>≤</mml:mo> <mml:msub><mml:mtext>dim</mml:mtext> <mml:mtext>A</mml:mtext></mml:msub> <mml:mi>F</mml:mi></mml:mrow> </mml:math> that there is a function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϕ</mml:mi></mml:math> so that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϕ</mml:mi></mml:math> -Assouad dimension of <i>F</i> is equal to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> . We further show that the "upper" variant of the dimension is fully determined by the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϕ</mml:mi></mml:math> -Assouad dimension, and that homogeneous Moran sets are in a certain sense generic for these dimensions. Further, we study explicit examples of sets where the Assouad spectrum does not reach the Assouad dimension. We prove a precise formula for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϕ</mml:mi></mml:math> -Assouad dimensions for the boundary of Galton-Watson trees that correspond to a general class of stochastically self-similar sets, including Mandelbrot percolation. The proof of this result combines a sharp large deviations theorem for Galton-Watson processes with bounded offspring distribution and a general Borel-Cantelli-type lemma for infinite structures in random trees. Finally, we obtain results on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϕ</mml:mi></mml:math> -Assouad dimensions of overlapping self-similar sets and decreasing sequences with decreasing gaps.