Displaying similar documents to “The Lucas congruence for Stirling numbers of the second kind”

The 2-Sylow subgroups of the tame kernel of imaginary quadratic fields

Hourong Qin (1995)

Acta Arithmetica

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1. Introduction. Let F be a number field and O F the ring of its integers. Many results are known about the group K ₂ O F , the tame kernel of F. In particular, many authors have investigated the 2-Sylow subgroup of K ₂ O F . As compared with real quadratic fields, the 2-Sylow subgroups of K ₂ O F for imaginary quadratic fields F are more difficult to deal with. The objective of this paper is to prove a few theorems on the structure of the 2-Sylow subgroups of K ₂ O F for imaginary quadratic fields F. In our Ph.D....

Shift spaces and attractors in noninvertible horseshoes

H. Bothe (1997)

Fundamenta Mathematicae

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As is well known, a horseshoe map, i.e. a special injective reimbedding of the unit square I 2 in ℝ 2 (or more generally, of the cube I m in ℝ m ) as considered first by S. Smale [5], defines a shift dynamics on the maximal invariant subset of I 2 (or I m ). It is shown that this remains true almost surely for noninjective maps provided the contraction rate of the mapping in the stable direction is sufficiently strong, and bounds for this rate are given.

Growth of the product ∏ j = 1 n ( 1 - x a j )

J. P. Bell, P. B. Borwein, L. B. Richmond (1998)

Acta Arithmetica

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We estimate the maximum of ∏ j = 1 n | 1 - x a j | on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when a j is j k or when a j is a quadratic in j that takes on positive integer values, the maximum grows as exp(cn), where c is a positive constant. This complements results of Sudler and Wright that show exponential growth when a j is j.    In contrast we show, under fairly general conditions, that the maximum is less than 2 n / n r , where r is an arbitrary positive number. One consequence...

A complement to the theory of equivariant finiteness obstructions

Paweł Andrzejewski (1996)

Fundamenta Mathematicae

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It is known ([1], [2]) that a construction of equivariant finiteness obstructions leads to a family w α H ( X ) of elements of the groups K 0 ( ℤ [ π 0 ( W H ( X ) ) α * ] ) . We prove that every family w α H of elements of the groups K 0 ( ℤ [ π 0 ( W H ( X ) ) α * ] ) can be realized as the family of equivariant finiteness obstructions w α H ( X ) of an appropriate finitely dominated G-complex X. As an application of this result we show the natural equivalence of the geometric construction of equivariant finiteness obstruction ([5], [6]) and equivariant generalization of Wall’s...

Length of continued fractions in principal quadratic fields

Guillaume Grisel (1998)

Acta Arithmetica

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Let d ≥ 2 be a square-free integer and for all n ≥ 0, let l ( ( √ d ) 2 n + 1 ) be the length of the continued fraction expansion of ( √ d ) 2 n + 1 . If ℚ(√d) is a principal quadratic field, then under a condition on the fundamental unit of ℤ[√d] we prove that there exist constants C₁ and C₂ such that C ₁ ( √ d ) 2 n + 1 ≥ l ( ( √ d ) 2 n + 1 ) ≥ C ₂ ( √ d ) 2 n + 1 for all large n. This is a generalization of a theorem of S. Chowla and S. S. Pillai [2] and an improvement in a particular case of a theorem of [6].

An entropy for ℤ 2 -actions with finite entropy generators

W. Geller, M. Pollicott (1998)

Fundamenta Mathematicae

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We study a definition of entropy for ℤ + × ℤ + -actions (or ℤ 2 -actions) due to S. Friedland. Unlike the more traditional definition, this is better suited for actions whose generators have finite entropy as single transformations. We compute its value in several examples. In particular, we settle a conjecture of Friedland [2].

How to recognize a true Σ^0_3 set

Etienne Matheron (1998)

Fundamenta Mathematicae

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Let X be a Polish space, and let ( A p ) p ∈ ω be a sequence of G δ hereditary subsets of K(X) (the space of compact subsets of X). We give a general criterion which allows one to decide whether ∪ p ∈ ω A p is a true ∑ 3 0 subset of K(X). We apply this criterion to show that several natural families of thin sets from harmonic analysis are true ∑ 3 0 .

Distribution of lattice points on hyperbolic surfaces

Vsevolod F. Lev (1996)

Acta Arithmetica

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Let two lattices Λ ' , Λ ' ' ⊂ ℝ s have the same number of points on each hyperbolic surface | x ₁ . . . x s | = C . We investigate the case when Λ’, Λ” are sublattices of ℤ s of the same prime index and show that then Λ’ and Λ” must coincide up to renumbering the coordinate axes and changing their directions.

On strong uniform distribution, II. The infinite-dimensional case

Y. Lacroix (1998)

Acta Arithmetica

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We construct infinite-dimensional chains that are L¹ good for almost sure convergence, which settles a question raised in this journal [N]. We give some conditions for a coprime generated chain to be bad for L² or L ∞ , using the entropy method. It follows that such a chain with positive lower density is bad for L ∞ . There also exist such bad chains with zero density.

The Iwasawa λ-invariants of ℤₚ-extensions of real quadratic fields

Takashi Fukuda, Hisao Taya (1995)

Acta Arithmetica

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1. Introduction. Let k be a totally real number field. Let p be a fixed prime number and ℤₚ the ring of all p-adic integers. We denote by λ=λₚ(k), μ=μₚ(k) and ν=νₚ(k) the Iwasawa invariants of the cyclotomic ℤₚ-extension k ∞ of k for p (cf. [10]). Then Greenberg’s conjecture states that both λₚ(k) and μₚ(k) always vanish (cf. [8]). In other words, the order of the p-primary part of the ideal class group of kₙ remains bounded as n tends to infinity, where kₙ is the nth layer of k ∞ / k . We know...