  We introduce a new dimension spectrum motivated by the Assouad dimension; a
familiar notion of dimension which, for a given metric space, returns the
minimal exponent $\alpha\geq 0$ such that for any pair of scales $0<r<R$, any
ball of radius $R$ may be covered by a constant times $(R/r)^\alpha$ balls of
radius $r$. To each $\theta \in (0,1)$, we associate the appropriate analogue
of the Assouad dimension with the restriction that the two scales $r$ and $R$
used in the definition satisfy $\log R/\log r = \theta$. The resulting
`dimension spectrum' (as a function of $\theta$) thus gives finer geometric
information regarding the scaling structure of the space and, in some precise
sense, interpolates between the upper box dimension and the Assouad dimension.
This latter point is particularly useful because the spectrum is generally
better behaved than the Assouad dimension. We also consider the corresponding
`lower spectrum', motivated by the lower dimension, which acts as a dual to the
Assouad spectrum.
  We conduct a detailed study of these dimension spectra; including analytic,
geometric, and measureability properties. We also compute the spectra
explicitly for some common examples of fractals including decreasing sequences
with decreasing gaps and spirals with sub-exponential and monotonic winding. We
also give several applications of our results, including: dimension distortion
estimates under bi-H\"older maps for Assouad dimension and the provision of new
bi-Lipschitz invariants.
