Summary
It is well-known that almost every number in [0, 1] is normal in base 2, in the sense of Lebesgue measure. Kahane and Salem asked whether the same is true with respect to any Borel measure whose Fourier-Stieltjes coefficients vanish at infinity — in other words, whether the set of non-normal numbers is a set of uniqueness in the wide sense. We show that this is not the case. In fact, we give “best-possible” conditions on the rate of decay of\(\hat \mu (n)\) in order that μ-almost every number be normal. The techniques include, on the one hand, probability measures with respect to which the binary digits in [0, 1] are independent only by blocks, rather than individually, and on the other hand, the strong law of large numbers for weakly correlated random variables.
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This work was partially supported by an NSF Graduate Fellowship, NSF Grant MCS-82-01602, and an AMS Research Fellowship.
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Lyons, R. The measure of non-normal sets. Invent Math 83, 605–616 (1986). https://doi.org/10.1007/BF01394426
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DOI: https://doi.org/10.1007/BF01394426