Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ {0,1} by $dₙ(x) := ∑_{i=1}^{n} 1_{E} ({2^{i-1}x}) (mod 2)$, where t is the fractional part of t. Then x is called a normal number mod 2 with respect to E if $N^{-1} ∑_{n=1}^{N} dₙ(x)$ converges to 1/2. It is shown that for any interval E ≠(1/6, 5/6) a.e. x is a normal number mod 2 with respect to E. For E = (1/6, 5/6) it is proved that $N^{-1} ∑_{n=1}^{N} dₙ(x)$ converges a.e. and the limit equals 1/3 or 2/3 depending on x.
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We show that semisimple actions of l.c.s.c. Abelian groups and cocycles with values in such groups can be used to build new examples of semisimple automorphisms (ℤ-actions) which are relatively weakly mixing extensions of irrational rotations.
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