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Partial densities on the group of integers

Harald Niederreiter, Norris Sookoo (2000)

Archivum Mathematicum

Conditions are obtained under which a partial density on the group of integers with the discrete topology can be extended to a density.

Periodic Jacobi-Perron expansions associated with a unit

Brigitte Adam, Georges Rhin (2011)

Journal de Théorie des Nombres de Bordeaux

We prove that, for any unit ϵ in a real number field K of degree n + 1 , there exits only a finite number of n-tuples in  K n which have a purely periodic expansion by the Jacobi-Perron algorithm. This generalizes the case of continued fractions for n = 1 . For n = 2 we give an explicit algorithm to compute all these pairs.

Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences

M. Blümlinger, N. Obata (1991)

Annales de l'institut Fourier

We are interested in permutations preserving certain distribution properties of sequences. In particular we consider μ -uniformly distributed sequences on a compact metric space X , 0-1 sequences with densities, and Cesàro summable bounded sequences. It is shown that the maximal subgroups, respectively subsemigroups, of A u t ( N ) leaving any of the above spaces invariant coincide. A subgroup of these permutation groups, which can be determined explicitly, is the Lévy group 𝒢 . We show that 𝒢 is big in the...

Piecewise convex transformations with no finite invariant measure

Tomasz Komorowski (1991)

Annales Polonici Mathematici

 Abstract. The paper concerns the problem of the existence of a finite invariant absolutely continuous measure for piecewise C 2 -regular and convex transformations T: [0, l]→[0,1]. We show that in the case when T’(0) = 1 and T"(0) exists T does not admit such a measure. This result is complementary to the ones contained in [3] and [5].

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