  We unite two themes in dyadic analysis and number theory by studying an
analogue of the failure of the Hasse principle in harmonic analysis.
Explicitly, we construct an explicit family of measures on the real line that
are $p$-adic and $q$-adic doubling for any distinct primes $p$ and $q$, yet not
doubling, and we apply these results to show analogous statements about the
reverse H\"older and Muckenhoupt $A_p$ classes of weights. The proofs involve a
delicate interplay among several geometric and number theoretic properties.
