On the sums
J. C. Peral (1995)
Acta Arithmetica
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J. C. Peral (1995)
Acta Arithmetica
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Maohua Le (1997)
Acta Arithmetica
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1. Introduction. Let ℤ, ℕ, ℚ be the sets of integers, positive integers and rational numbers respectively. In [7], Ribenboim proved that the equation (1) , x,y,m,n ∈ ℕ, x > 1, n > m ≥ 1, has no solution (x,y,m,n) with 2|x and (1) has only finitely many solutions (x,y,m,n) with 2∤x. Moreover, all solutions of (1) with 2∤x satisfy max(x,m,n) < C, where C is an effectively computable constant. In this paper we completely determine all solutions of (1) as follows. Theorem....
K. Győry (1998)
Acta Arithmetica
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Michał Sierakowski (1999)
Fundamenta Mathematicae
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E. C. Zeeman [2] described the behaviour of the iterates of the difference equation , n ≥ k, R,Q polynomials in the case and , positive, α nonnegative. We generalize his results as well as those of Beukers and Cushman on the existence of an invariant measure in the case when R,Q are affine and k = 1. We prove that the totally invariant set remains residual when the coefficients vary.
Nobuhiro Terai (1993)
Acta Arithmetica
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Imre Z. Ruzsa (1993)
Acta Arithmetica
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Menachem Kojman (2000)
Fundamenta Mathematicae
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A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function ϱ is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then ϱ is used to give a necessary and sufficient condition for countable convexity of closed sets. Theorem. Suppose that S is a closed subset of a...
Ziv Shami, Saharon Shelah (1999)
Fundamenta Mathematicae
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In a countable superstable NDOP theory, the existence of a rigid -saturated model implies the existence of rigid -saturated models of power λ for every .
Paul Erdös, Steve Jackson, R. Mauldin (1994)
Fundamenta Mathematicae
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We consider a set, L, of lines in and a partition of L into some number of sets: . We seek a corresponding partition such that each line l in meets the set in a set whose cardinality has some fixed bound, . We determine equivalences between the bounds on the size of the continuum, , and some relationships between p, and .
Tomasz Łuczak, Tomasz Schoen (2000)
Acta Arithmetica
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Martin Helm (1993)
Acta Arithmetica
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Anzelm Iwanik (1998)
Fundamenta Mathematicae
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Let α be an ergodic rotation of the d-torus . For any piecewise smooth function with sufficiently regular pieces the unitary operator Vh(x) = exp(2π if(x))h(x + α) acting on is shown to have a continuous non-Dirichlet spectrum if the gradient of f has nonzero integral. In particular, the resulting skew product must be ergodic. If in addition α is sufficiently well approximated by rational vectors and f is represented by a linear function with noninteger coefficients then the spectrum...
Tamás Keleti (1998)
Fundamenta Mathematicae
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We investigate some problems of the following type: For which sets H is it true that if f is in a given class ℱ of periodic functions and the difference functions are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by ℌ(ℱ,G), that is, , we try to characterize ℌ(ℱ,G) for some interesting classes of functions ℱ ⊃ G. We study classes of measurable functions on the circle group that are invariant for changes on null-sets...