Hausdorff’s forgotten proof that almost all numbers are normal

Weitz, Edmund

doi:10.1007/s00591-021-00303-w

Cited by 1 publication

(2 citation statements)

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“…This reframing allows us to ask and answer questions about these polynomials measure-theoretically. For example, we will use Borel's theorem that every number is normal, regardless of base [Wei21].…”

Section: Proof the Proportion Of Degreementioning

confidence: 99%

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Irreducibility over the Max-Min Semiring

Baily¹,

Dell²,

L.³

et al. 2021

Preprint

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For sets A, B ⊂ N, their sumset is A + B := {a + b : a ∈ A, b ∈ B}. If we cannot write a set C as C = A + B with |A|, |B|≥ 2, then we say that C is irreducible. The question of whether a given set C is irreducible arises naturally in additive combinatorics. Equivalently, we can formulate this question as one about the irreducibility of boolean polynomials, which has been discussed in previous work by K. H. Kim and F. W. Roush (2005) and Y. Shitov ( 2014). We prove results about the irreducibility of polynomials and power series over the max-min semiring, a natural generalization of the boolean polynomials.We use combinatorial and probabilistic methods to prove that almost all polynomials are irreducible over the max-min semiring, generalizing work of Y. Shitov (2014) and proving a 2011 conjecture by D. L. Applegate, M. Le Brun, and N. J. A. Sloane. Furthermore, we use measure-theoretic methods and apply Borel's result on normal numbers to prove that almost all power series are asymptotically irreducible over the max-min semiring.

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Section: Proof the Proportion Of Degreementioning

confidence: 99%

“…In Section 3, we prove Theorem 1.9 by partitioning the set of reducible elements of B b [[x]] into subcollections and bounding the size of each. This is done via applications of the Borel-Cantelli Lemma and the result that almost all numbers are normal in every base [Wei21].…”

Section: Introductionmentioning

confidence: 99%