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Remarks on Steinhaus’ property and ratio sets of sets of positive integers

Tibor Šalát (2000)

Czechoslovak Mathematical Journal

This paper is closely related to an earlier paper of the author and W. Narkiewicz (cf. [7]) and to some papers concerning ratio sets of positive integers (cf. [4], [5], [12], [13], [14]). The paper contains some new results completing results of the mentioned papers. Among other things a characterization of the Steinhaus property of sets of positive integers is given here by using the concept of ratio sets of positive integers.

Représentation des entiers naturels et suites uniformément équiréparties

Jean Coquet (1982)

Annales de l'institut Fourier

s ( n ) désigne la somme des chiffres de l’entier n en base q et σ α ( n ) la somme des chiffres de n associée au développement de α en fraction continue. Dans un article paru aux Annales de l’Institut Fourier (31 (1981), 1–15), Coquet, Rhin et Toffin montrent que, lorsque x ou y est irrationnel, la suite x s + y σ α est équirépartie modulo 1. On précise ici que l’équirépartition est uniforme.

Run-length function of the Bolyai-Rényi expansion of real numbers

Rao Li, Fan Lü, Li Zhou (2024)

Czechoslovak Mathematical Journal

By iterating the Bolyai-Rényi transformation T ( x ) = ( x + 1 ) 2 ( mod 1 ) , almost every real number x [ 0 , 1 ) can be expanded as a continued radical expression x = - 1 + x 1 + x 2 + + x n + with digits x n { 0 , 1 , 2 } for all n . For any real number x [ 0 , 1 ) and digit i { 0 , 1 , 2 } , let r n ( x , i ) be the maximal length of consecutive i ’s in the first n digits of the Bolyai-Rényi expansion of x . We study the asymptotic behavior of the run-length function r n ( x , i ) . We prove that for any digit i { 0 , 1 , 2 } , the Lebesgue measure of the set D ( i ) = x [ 0 , 1 ) : lim n r n ( x , i ) log n = 1 log θ i is 1 , where θ i = 1 + 4 i + 1 . We also obtain that the level set E α ( i ) = x [ 0 , 1 ) : lim n r n ( x , i ) log n = α is of full Hausdorff dimension...

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