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Displaying similar documents to “The catenary degree of Krull monoids I”

Factorization in Krull monoids with infinite class group

Florian Kainrath (1999)

Colloquium Mathematicae

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Let H be a Krull monoid with infinite class group and such that each divisor class of H contains a prime divisor. We show that for each finite set L of integers ≥2 there exists some h ∈ H such that the following are equivalent: (i) h has a representation h = u 1 · . . . · u k for some irreducible elements u i , (ii) k ∈ L.

On ergodicity of some cylinder flows

Krzysztof Frączek (2000)

Fundamenta Mathematicae

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We study ergodicity of cylinder flows of the form    T f : T × T × , T f ( x , y ) = ( x + α , y + f ( x ) ) , where f : T is a measurable cocycle with zero integral. We show a new class of smooth ergodic cocycles. Let k be a natural number and let f be a function such that D k f is piecewise absolutely continuous (but not continuous) with zero sum of jumps. We show that if the points of discontinuity of D k f have some good properties, then T f is ergodic. Moreover, there exists ε f > 0 such that if v : T is a function with zero integral such that D k v is of bounded...

Parametrized Cichoń's diagram and small sets

Janusz Pawlikowski, Ireneusz Recław (1995)

Fundamenta Mathematicae

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We parametrize Cichoń’s diagram and show how cardinals from Cichoń’s diagram yield classes of small sets of reals. For instance, we show that there exist subsets N and M of w w × 2 w and continuous functions e , f : w w w w such that  • N is G δ and N x : x w w , the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of 2 w ;  • M is F σ and M x : x w w is a basis for the ideal of meager subsets of 2 w ;  • x , y N e ( x ) N y M x M f ( y ) . From this we derive that for a separable metric space X,  •if for all Borel (resp. G δ ) sets...

On B 2 k -sequences

Martin Helm (1993)

Acta Arithmetica

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Introduction. An old conjecture of P. Erdős repeated many times with a prize offer states that the counting function A(n) of a B r -sequence A satisfies l i m i n f n ( A ( n ) / ( n 1 / r ) ) = 0 . The conjecture was proved for r=2 by P. Erdős himself (see [5]) and in the cases r=4 and r=6 by J. C. M. Nash in [4] and by Xing-De Jia in [2] respectively. A very interesting proof of the conjecture in the case of all even r=2k by Xing-De Jia is to appear in the Journal of Number Theory [3]. Here we present a different, very short proof...

Analytic determinacy and 0# A forcing-free proof of Harrington’s theorem

Ramez Sami (1999)

Fundamenta Mathematicae

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We prove the following theorem: Given a⊆ω and 1 α < ω 1 C K , if for some η < 1 and all u ∈ WO of length η, a is Σ α 0 ( u ) , then a is Σ α 0 . We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: Σ 1 1 -Turing-determinacy implies the existence of 0 .

Bing maps and finite-dimensional maps

Michael Levin (1996)

Fundamenta Mathematicae

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Let X and Y be compacta and let f:X → Y be a k-dimensional map. In [5] Pasynkov stated that if Y is finite-dimensional then there exists a map g : X 𝕀 k such that dim (f × g) = 0. The problem that we deal with in this note is whether or not the restriction on the dimension of Y in the Pasynkov theorem can be omitted. This problem is still open.  Without assuming that Y is finite-dimensional Sternfeld [6] proved that there exists a map g : X 𝕀 k such that dim (f × g) = 1. We improve this result of Sternfeld...

Inessentiality with respect to subspaces

Michael Levin (1995)

Fundamenta Mathematicae

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Let X be a compactum and let A = ( A i , B i ) : i = 1 , 2 , . . . be a countable family of pairs of disjoint subsets of X. Then A is said to be essential on Y ⊂ X if for every closed F i separating A i and B i the intersection ( F i ) Y is not empty. So A is inessential on Y if there exist closed F i separating A i and B i such that F i does not intersect Y. Properties of inessentiality are studied and applied to prove:  Theorem. For every countable family of pairs of disjoint open subsets of a compactum X there exists an open set G ∩ X on...

A note on Tsirelson type ideals

Boban Veličković (1999)

Fundamenta Mathematicae

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Using Tsirelson’s well-known example of a Banach space which does not contain a copy of c 0 or l p , for p ≥ 1, we construct a simple Borel ideal I T such that the Borel cardinalities of the quotient spaces P ( ) / I T and P ( ) / I 0 are incomparable, where I 0 is the summable ideal of all sets A ⊆ ℕ such that n A 1 / ( n + 1 ) < . This disproves a “trichotomy” conjecture for Borel ideals proposed by Kechris and Mazur.

On the restricted Waring problem over 2 n [ t ]

Luis Gallardo (2000)

Acta Arithmetica

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1. Introduction. The Waring problem for polynomial cubes over a finite field F of characteristic 2 consists in finding the minimal integer m ≥ 0 such that every sum of cubes in F[t] is a sum of m cubes. It is known that for F distinct from ₂, ₄, 16 , each polynomial in F[t] is a sum of three cubes of polynomials (see [3]). If a polynomial P ∈ F[t] is a sum of n cubes of polynomials in F[t] such that each cube A³ appearing in the decomposition has degree < deg(P)+3, we say that P is...

A problem of Galambos on Engel expansions

Jun Wu (2000)

Acta Arithmetica

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1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) x = 1 / d ( x ) + 1 / ( d ( x ) d ( x ) ) + . . . + 1 / ( d ( x ) d ( x ) . . . d n ( x ) ) + . . . , where d j ( x ) , j 1 is a sequence of positive integers satisfying d₁(x) ≥ 2 and d j + 1 ( x ) d j ( x ) for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) l i m n d n 1 / n ( x ) = e . He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. d i m H x ( 0 , 1 ] : ( 2 ) f a i l s = 1 . We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and d i m H to denote...