Growth Rate for Beta-expansions
نویسندگان
چکیده
Let β > 1 and let m > β be an integer. Each x ∈ Iβ := [0, m−1 β−1 ] can be represented in the form x = ∞ ∑ k=1 εkβ −k, where εk ∈ {0, 1, . . . , m − 1} for all k (a β-expansion of x). It is known that a.e. x ∈ Iβ has a continuum of distinct β-expansions. In this paper we prove that if β is a Pisot number, then for a.e. x this continuum has one and the same growth rate. We also link this rate to the Lebesgue-generic local dimension for the Bernoulli convolution parametrized by β. When β < 1+ √ 5 2 , we show that the set of β-expansions grows exponentially for every internal x.
منابع مشابه
تحلیل و آزمون عدم تقارن در رفتار سیاستگذاری پولی بانک مرکزی
According to Taylor (1993) rule, the monetary authority responds to deviations of output and of inflation from their targets through nominal interest rate fluctuations regarded as policy instrument. Another specification that has received considerable attention is that policymakers may have asymmetric preferences with regard to their objectives during recessions and expansions. Since according ...
متن کاملStability of $g$-Frame Expansions
In this paper we investigate the stability of one-sided perturbation to g-frame expansions. We show that if $Lambda$ is a g-frame of a Hilbert space $mathcal{H}$, $Lambda_{i}^{a}=Lambda_{i}+Theta_{i}$ where $Theta_{i} in mathcal{L}(mathcal{H},mathcal{H}_{i})$, and $widetilde{f}=sum_{i in J}Lambda_{i}^{star}widetilde{Lambda}_{i}^{a}f$, $widehat{f}=sum_{i in J}(Lambda_{i}^{a})^{star}widetilde{Lam...
متن کاملar X iv : 0 90 2 . 04 88 v 2 [ m at h . N T ] 1 1 M ay 2 00 9 GROWTH RATE FOR BETA - EXPANSIONS
Let β > 1 and let m > β be an integer. Each x ∈ Iβ := [0, m−1 β−1 ] can be represented in the form x = ∞ ∑ k=1 εkβ , where εk ∈ {0, 1, . . . ,m − 1} for all k (a β-expansion of x). It is known that a.e. x ∈ Iβ has a continuum of distinct β-expansions. In this paper we prove that if β is a Pisot number, then for a.e. x, this continuum has one and the same growth rate. We also link this rate to t...
متن کاملar X iv : 0 90 2 . 04 88 v 3 [ m at h . N T ] 1 7 A ug 2 00 9 GROWTH RATE FOR BETA - EXPANSIONS
Let β > 1 and let m > β be an integer. Each x ∈ Iβ := [0, m−1 β−1 ] can be represented in the form x = ∞ ∑ k=1 εkβ , where εk ∈ {0, 1, . . . ,m − 1} for all k (a β-expansion of x). It is known that a.e. x ∈ Iβ has a continuum of distinct β-expansions. In this paper we prove that if β is a Pisot number, then for a.e. x this continuum has one and the same growth rate. We also link this rate to th...
متن کاملNonharmonic Gabor Expansions
We consider Gabor systems generated by a Gaussian function and prove certain classical results of Paley and Wiener on nonharmonic Fourier series of complex exponentials for the Gabor expansion. In particular, we prove a version of Plancherel-Po ́lya theorem for entire functions with finite order of growth and use the Hadamard factorization theorem to study regularity, exactness and deficienc...
متن کامل