Submission + - Turing's unpublished algorithm for normal numbers (dc.uba.ar)
ferespo writes: A group of argentine scientists have given an almost full proof for an algorithm by Alan Turing, hand written by himself on the reverse of the sheets of his original paper about what would later be known as "Turing machines"
"In an unpublished manuscript, Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he gave an algorithm for computing instances in this set. We complete his manuscript by giving full proofs and correcting minor errors. While doing this, we recreate Turing's ideas as accurately as possible. One of his original lemmas remained unproved, but we have replaced it with a weaker lemma that still allows us to maintain Turing's proof idea and obtain his result."
"In an unpublished manuscript, Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he gave an algorithm for computing instances in this set. We complete his manuscript by giving full proofs and correcting minor errors. While doing this, we recreate Turing's ideas as accurately as possible. One of his original lemmas remained unproved, but we have replaced it with a weaker lemma that still allows us to maintain Turing's proof idea and obtain his result."