Rarified sums of the Thue-Morse sequence

Drmota, Michael; Skałba, Mariusz

doi:10.1090/s0002-9947-99-02277-1

Cited by 16 publications

(6 citation statements)

References 8 publications

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“…Let us observe [DS2] that the assertion that ψ p,j (x) has no zero is more or less equivalent to the Newman-type inequality S p,j (n) > 0 for almost all n (or S p,j (n) < 0 for almost all n)…”

Section: The Rarefaction Phenomenonmentioning

confidence: 99%

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On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon

Gazeau¹,

Verger-Gaugry²

2008

J. Théor. Nombres Bordeaux

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On explore le spectre d'un peigne de Dirac pondéré supporté par le quasicristal de Thue-Morse, et on le caractérise à un ensemble de mesure nulle près, au moyen de la Conjecture de Bombieri-Taylor, pour les pics de Bragg, et d'une autre conjecture que l'on appelle Conjecture de Aubry-Godrèche-Luck, pour la composante singulière continue. La décomposition de la transformée de Fourier du peigne de Dirac pondéré est obtenue dans le cadre de la théorie des distributions tempérées. Nous montrons que l'asymptotique de l'arithmetique des sommes p-raréfiées de Thue-Morse (Dumont; Goldstein, Kelly and Speer; Grabner; Drmota and Skalba,...), précisément les fonctions fractales des sommes de chiffres, jouent un rôle fondamental dans la description de la composante singulière continue du spectre, combinées à des résultats classiques sur les produits de Riesz de Peyrière et de M. Queffélec. Les lois d'échelle dominantes des suites de mesures approximantes sont contrôlées sur une partie de la composante singulière continue par certaines inégalités dans lesquelles le nombre de classes de diviseurs et le régulateur de corps quadratiques réels interviennent.

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Section: The Rarefaction Phenomenonmentioning

confidence: 99%

“…where "almost all" means "all but finitely many". In [DS2] Drmota and Skalba prove (here p denotes an arbitrary integer ≥ 3 and N ≥ 1 an integer)…”

Section: The Rarefaction Phenomenonmentioning

confidence: 99%

On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon

Gazeau¹,

Verger-Gaugry²

2008

J. Théor. Nombres Bordeaux

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“…Morgenbesser, Shallit and Stoll [18] proved that for p ≥ 1, min{n : t(pn) = 1} ≤ p + 4, and this becomes sharp for p = 2 2r −1 for r ≥ 1 (Note that 3 = 2 2•1 −1 is exactly of that form). A huge literature is nowadays available for classes of arithmetic progressions where such Newman-type phenomena exist and many generalizations have been considered so far (see [3,4,8,10,12,22] and the references given therein). Still, a full classification is not yet at our disposal.…”

Section: Introductionmentioning

confidence: 99%

Thue–Morse Along Two Polynomial Subsequences

Stoll

2015

Lecture Notes in Computer Science

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The aim of the present article is twofold. We first give a survey on recent developments on the distribution of symbols in polynomial subsequences of the Thue-Morse sequence t = (t(n)) n≥0 by highlighting effective results. Secondly, we give explicit bounds on min{n : (t(pn), t(qn)) = (ε1, ε2)}, for odd integers p, q, and on min{n : (t(n h 1), t(n h 2)) = (ε1, ε2)} where h1, h2 ≥ 1, and (ε1, ε2) is one of (0, 0), (0, 1), (1, 0), (1, 1).

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“…Rarified sums (or p-rarified sums, the term is due to [7]) of a sequence (t n ) n 0 are the sums of initial terms of the subsequence (t pn ) n 0 (the rarefaction step p is supposed to be a prime number in this paper). The problem of estimating the speed of growth of these sums has been studied in [8], [5], [9], [10], [11].…”

Section: Introductionmentioning

confidence: 99%