On the sets of uniqueness of a distribution function of {ξ(p/q)<sup>n</sup>}

Adhikari, Sukumar Das; Rath, Purusottam; Saradha, N.

doi:10.4064/aa119-4-1

Cited by 9 publications

(13 citation statements)

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“…The subject of distribution of powers of a number modulo 1 is classical and goes back to [10], [11], [17][18][19]. There are, however, many recent publications [1][2][3][4][5][6][7], [20], [21] on this subject too. We will compare our results with some of these in the next section.…”

Section: Theorem 13mentioning

confidence: 99%

On the powers of 3/2 and other rational numbers

Dubickas¹

2008

Mathematische Nachrichten

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we prove several results about subsets of the interval [0, 1) which does or does not contain all the fractional parts {ξ(p/q) n }, n = 0, 1, 2, . . . , for certain non-zero real number ξ. We show, for instance, that there are no real ξ for which the union of two intervals [8/39, 18/39] ∪ [21/39, 31/39] contains the set {ξ(3/2) n }, n ∈ N. The most important aspect of this result is that the total length of both intervals 20/39 is greater than 1/2: the same result as above for [0, 1/2) would imply that there are no Mahler's Z-numbers which the best known unsolved problem in this area. On the other hand, it is shown that there are infinitely many ξ for which {ξ(3/2) n } ∈ (5/48, 43/48) for each integer n ≥ 0. We also give simpler proofs of few recent results in this area.

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Section: Theorem 13mentioning

confidence: 99%

On the powers of 3/2 and other rational numbers

Dubickas¹

2008

Mathematische Nachrichten

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“…., where b 2 is a positive integer and / ∈ Q, have been considered in [14]. The case when = 0 and b > 1 is a rational non-integer number was considered in [3,5,13,23,26,27,36]. In [20], b > 1 is allowed to be algebraic, whereas the interval constructions of [4,37] allow b to be transcendental.…”

Section: Corollarymentioning

confidence: 99%

On a sequence related to that of Thue–Morse and its applications

Dubickas

2007

Discrete Mathematics

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It is known that the sequence 1, of lengths of blocks of identical symbols in the Thue-Morse sequence has several extremal properties among all non-periodic sequences of the symbols 1 and 2. Its generatingIn terms of combinatorics on words, for any given x ∈ (0, 1) and > 0, we prove that every non-periodic word of an alphabet {1, 2} has a suffix s whose generating function S(x) satisfies the inequality xS(−x) > 1−W (−x)− . Using this, we prove several bounds for the largest and the smallest limit points of the sequence of fractional parts { b n }, n = 0, 1, 2, . . ., where b < − 1 is a negative rational number and is a real number. Our results show, for example, that, for any real number = 0, the sequence of fractional parts { (−3/2) n }, n = 0, 1, 2, . . ., has a limit point greater than 0.466452. Furthermore, for each integer b −2 and each real number / ∈ Q, we prove that lim inf n→∞ { b n } ∞ k=1 (1−|b| −(2 k +(−1) k−1 )/3 ) and show that this inequality is sharp.

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“…. , and some related problems were considered earlier by Tijdeman [10], Pollington [11], and in the recent years in [12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

“…It is clear that the smaller the set A, the harder the problem. For some algebraic values of α and intervals A this problem was considered in [12][13][14] and [16][17][18]. It turns out that, for some algebraic numbers α > 1, for example, α = 5/2, there is ξ = 0 such that all fractional parts {ξα n }, n = 0, 1, 2, .…”

Section: Introductionmentioning

confidence: 99%