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1998 | 130 | 3 | 199-212
Tytuł artykułu

On the growth of averaged Weyl sums for rigid rotations

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Języki publikacji
EN
Abstrakty
Let ω ∈ ℝ╲ℚ and $f ∈ L^2(𝕊^1)$ of zero average. We study the asymptotic behaviour of the Weyl sums $S(m,ω)f(x) = ∑^{m-1}_{k=0} f(x + kω)$ and their averages $Ŝ(m,ω)f(x) = {1/m}∑^m_{j=1} S(j,ω)f(x)$, in the $L^2$-norm. In particular, for a suitable class of Liouville rotation numbers ω ∈ ℝ╲ℚ, we are able to construct examples of functions $f ∈ H^{s}𝕊^{1}$, s > 0, such that, for all ε > 0, ||Ŝ(m,ω)f||_2 ≥ C_{ε}m^{1/(1+s)-ε}$ as m → ∞. We show in addition that, for all $f ∈ H^{s}𝕊^{1}$, $lim inf m^{-1/(1+s)} (log m)^{-1/2} ||Ŝ(m,ω)f||_2 < ∞$ for all ω ∈ ℝ╲ℚ.
Słowa kluczowe
Czasopismo
Rocznik
Tom
130
Numer
3
Strony
199-212
Opis fizyczny
Daty
wydano
1998
otrzymano
1996-09-05
poprawiono
1998-01-12
Twórcy
  • UFR de Mathématiques et URA GAT, Université des Sciences et Technologies de Lille, 59655 Villeneuve d'Ascq Cedex, France, debievre@gat.univ-lille1.fr
autor
Bibliografia
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  • [Be] J. Bellissard, Stability and instability in quantum mechanics, in: Trends and Developments in the 80's (Bielefeld, 1982//1983), S. Albeverio and P. Blanchard (eds.), World Sci., 1985, 1-106.
  • [CMPS] D. Crisp, W. Moran, A. Pollington and P. Shiue, Substitution invariant cutting sequences, J. Théor. Nombres Bordeaux 5 (1993), 123-137.
  • [DBF] S. De Bièvre and G. Forni, Transport properties of kicked and quasi-periodic Hamiltonians, J. Statist. Phys. (1998), to appear.
  • [D] J. M. Dumont, Summation formulae for substitutions on a finite alphabet, in: Number Theory and Physics, J. M. Luck, P. Moursa and M. Waldschmidt (eds.), Lecture Notes in Phys. 47, Springer, 1990, 185-194.
  • [G] G. Gallavotti, Elements of Classical Mechanics, Texts Monographs Phys., Springer, 1983.
  • [GLV] C. Godrèche, J. M. Luck and F. Vallet, Quasiperiodicity and types of order a study in one dimension, J. Phys. A 20 (1987), 4483-4499.
  • [K] A. Khinchin, Continued Fractions, The Univ. of Chicago Press, Chicago and London, 1964.
  • [Ke] H. Kesten, On a conjecture by Erdős and Szüsz related to uniform distribution modulo 1, Acta Arith. 12 (1966), 193-212.
  • [KN] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York, 1974.
  • [L] P. Liardet, Regularities of distribution, Compositio Math. 61 (1987), 267-293.
  • [LV] P. Liardet and D. Volný, Sums of continuous and differentiable functions in dynamical systems, Israel J. Math. 98 (1997), 29-60.
  • [Pa] D. A. Pask, Skew products over the irrational rotation, ibid. 69 (1990), 65-74.
  • [P1] K. Petersen, Ergodic Theory, Cambridge Univ. Press, Cambridge, 1983.
  • [P2] K. Petersen, On a series of cosecants related to a problem in ergodic theory, Compositio Math. 26 (1973), 313-317.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv130i3p199bwm
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