Displaying similar documents to “Digital expansion of exponential sequences”

Besicovitch subsets of self-similar sets

Ji-Hua Ma, Zhi-Ying Wen, Jun Wu (2002)

Annales de l’institut Fourier

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Let E be a self-similar set with similarities ratio r j ( 0 ≤ j ≤ m - 1 ) and Hausdorff dimension s , let p → ( p 0 , p 1 ) ... p m - 1 be a probability vector. The Besicovitch-type subset of E is defined as E ( p → ) = x ∈ E : lim n → ∞ 1 n ∑ k = 1 n χ j ( x k ) = p j , 0 ≤ j ≤ m - 1 , where χ j is the indicator function of the set { j } . Let α = dim H ( E ( p → ) ) = dim P ( E ( p → ) ) = ∑ j = 0 m - 1 p j log p j ∑ j = 0 m - 1 p i log r j and g be a gauge function, then we prove in this paper:(i) If p → = ( r 0 s , r 1 s , ⋯ , r m - 1 s ) , then ℋ s ( E ( p → ) ) = ℋ s ( E ) , 𝒫 s ( E ( p → ) ) = 𝒫 s ( E ) , moreover both of ℋ s ( E ) and 𝒫 s ( E ) are finite positive;(ii) If p → is a positive probability vector other than ( r 0 s , r 1 s , ⋯ , r m - 1 s ) , then the gauge functions can be partitioned as follows ...

On the distribution of p α modulo one

Xiaodong Cao, Wenguang Zhai (1999)

Journal de théorie des nombres de Bordeaux

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In this paper, we give a new upper-bound for the discrepancy D ( N ) : = sup 0 ≤ γ ≤ 0 | ∑ p ≤ / N p α ≤ γ 1 - π ( N ) γ | for the sequence ( p α ) , when 5 / 3 ≤ α > 3 and α ≠ 2 .

On an estimate of Walfisz and Saltykov for an error term related to the Euler function

Y.-F. S. Pétermann (1998)

Journal de théorie des nombres de Bordeaux

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The technique developed by A. Walfisz in order to prove (in 1962) the estimate H ( x ) ≪ ( log x ) 2 / 3 ( log log x ) 4 / 3 for the error term H ( x ) = ∑ n ≤ x φ ( n ) n - 6 π 2 x related to the Euler function is extended. Moreover, the argument is simplified by exploiting works of A.I. Saltykov and of A.A. Karatsuba. It is noted in passing that the proof proposed by Saltykov in 1960 of H ( x ) ≪ ( log x ) 2 / 3 ( log log x ) 1 + ϵ is erroneous and once corrected “only” yields Walfisz’ result. The generalizations obtained can be applied to error terms related to various classical - and less classical -...

On the fractional parts of x / n and related sequences. II

Bahman Saffari, R. C. Vaughan (1977)

Annales de l'institut Fourier

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As promised in the first paper of this series (Ann. Inst. Fourier, 26-4 (1976), 115-131), these two articles deal with the asymptotic distribution of the fractional parts of x h ( x ) where h is an arithmetical function (namely h ( n ) = 1 / n , h ( n ) = log n , h ( n ) = 1 / log n ) and n is an integer (or a prime order) running over the interval [ y ( x ) , x ) ] . The results obtained are rather sharp, although one can improve on some of them at the cost of increased technicality. Number-theoretic applications will be given later on.

The Zeckendorf expansion of polynomial sequences

Michael Drmota, Wolfgang Steiner (2002)

Journal de théorie des nombres de Bordeaux

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In the first part of the paper we prove that the Zeckendorf sum-of-digits function s z ( n ) and similarly defined functions evaluated on polynomial sequences of positive integers or primes satisfy a central limit theorem. We also prove that the Zeckendorf expansion and the q -ary expansions of integers are asymptotically independent.