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 doubling for any finite set of primes, yet not doubling, and we apply these results to show analogous statements about the reverse Holder and Muckenhoupt $A_p$ classes of weights. The proofs involve a delicate interplay among several geometric and number theoretic properties.
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