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We introduce a new dimension spectrum motivated by the Assouad dimension; afamiliar notion of dimension which, for a given metric space, returns theminimal exponent $\alpha\geq 0$ such that for any pair of scales $0<r<R$, anyball of radius $R$ may be covered by a constant times $(R/r)^\alpha$ balls ofradius $r$. To each $\theta \in (0,1)$, we associate the appropriate analogueof 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 geometricinformation regarding the scaling structure of the space and, in some precisesense, interpolates between the upper box dimension and the Assouad dimension.This latter point is particularly useful because the spectrum is generallybetter behaved than the Assouad dimension. We also consider the corresponding`lower spectrum', motivated by the lower dimension, which acts as a dual to theAssouad spectrum.  We conduct a detailed study of these dimension spectra; including analytic,geometric, and measureability properties. We also compute the spectraexplicitly for some common examples of fractals including decreasing sequenceswith decreasing gaps and spirals with sub-exponential and monotonic winding. Wealso give several applications of our results, including: dimension distortionestimates under bi-H\"older maps for Assouad dimension and the provision of newbi-Lipschitz invariants.

New dimension spectra: finer information on scaling and homogeneity