On normal numbers mod $2$

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It is proved that a real-valued function

f (x) = \exp (π i χ_{I} (x))

, where I is an interval contained in [0,1), is not of the form

f (x) = \bar{q (2 x)} q (x)

with |q(x)|=1 a.e. if I has dyadic endpoints. A relation of this result to the uniform distribution mod 2 is also shown.

coboundary, metric density, normal number, uniform distribution

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