Given an arbitrary countable subgroup $σ_0$ of the torus, containing infinitely many rationals, we construct a strictly ergodic 0-1 Toeplitz flow with pure point spectrum equal to $σ_0$. For a large class of Toeplitz flows certain eigenvalues are induced by eigenvalues of the flow Y which can be seen along the aperiodic parts.
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Given a 0-1 sequence x in which both letters occur with density 1/2, do there exist arbitrarily long arithmetic progressions along which x reads 010101...? We answer the above negatively by showing that a certain regular triadic Toeplitz sequence does not have this property. On the other hand, we prove that if x is a generalized binary Morse sequence then each block can be read in x along some arithmetic progression.
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We construct strictly ergodic 0-1 Toeplitz flows with pure point spectrum and irrational eigenvalues. It is also shown that the property of being regular is not a measure-theoretic invariant for strictly ergodic Toeplitz flows.
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