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The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra

We consider the Assouad spectrum, introduced by Fraser and Yu, along with a natural variant that we call the `upper Assouad spectrum'. These spectra are designed to interpolate between the upper box-counting and Assouad dimensions. It is known that the Assouad spectrum approaches the upper box-counting dimension at the left hand side of its domain, but does not necessarily approach the Assouad dimension on the right. Here we show that it necessarily approaches the \emph{quasi-Assouad dimension} at the right hand side of its domain. We further show that the upper Assouad spectrum can be expressed in terms of the Assouad spectrum, thus motivating the definition used by Fraser-Yu. 
We also provide a large family of examples demonstrating new phenomena relating to the form of the Assouad spectrum. For example, we prove that it can be strictly concave, exhibit phase transitions of any order, and need not be piecewise differentiable.