Multiple Spectra of Bernoulli Convolutions

2016 ◽  
Vol 60 (1) ◽  
pp. 187-202 ◽  
Author(s):  
Jian-Lin Li ◽  
Dan Xing
Keyword(s):  
Finite Group ◽  
Spectral Measure ◽  
Bernoulli Convolution ◽  
Bernoulli Convolutions ◽  
Order Of Elements

AbstractLet μλ be the Bernoulli convolution associated with λ ∈ (0, 1). The well-known result of Jorgensen and Pedersen shows that if λ = 1/(2k) for some k ∈ ℕ, then μ1/(2k) is a spectral measure with spectrum Γ(1/(2k)). The recent research on the spectrality of μλ shows that μλ is a spectral measure only if λ = 1/(2k) for some k ∈ ℕ. Moreover, for certain odd integer p, the multiple set pΓ(1/(2k)) is also a spectrum for μ1/(2k). This is surprising because some spectra for the measure μ1/(2k) are thinning. In this paper we mainly characterize the number p that has the above property. By applying the properties of congruences and the order of elements in the finite group, we obtain several conditions on p such that pΓ(1/(2k)) is a spectrum for μ1/(2k).

scholarly journals On the Hausdorff Dimension of Bernoulli Convolutions

2018 ◽  
Vol 2020 (19) ◽  
pp. 6569-6595 ◽  
Author(s):  
Shigeki Akiyama ◽  
De-Jun Feng ◽  
Tom Kempton ◽  
Tomas Persson
Keyword(s):  
Hausdorff Dimension ◽  
Rate Of Convergence ◽  
Finite Time ◽  
Time Algorithm ◽  
Bernoulli Convolution ◽  
Bernoulli Convolutions ◽  
Explicit Rate ◽  
Products Of Matrices

Abstract We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution $\nu _{\beta }$ to arbitrary given accuracy whenever $\beta $ is algebraic. In particular, if the Garsia entropy $H(\beta )$ is not equal to $\log (\beta )$ then we have a finite time algorithm to determine whether or not $\operatorname{dim_H} (\nu _{\beta })=1$.

scholarly journals LEBESGUE STRUCTURE OF DISTRIBUTION OF GENERALIZED BERNOULLI CONVOLUTIONS OF SPECIAL TYPE

2020 ◽  
pp. 15-18
Author(s):  
O. Makarchuk ◽  
K. Salnik
Keyword(s):  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Special Kind ◽  
Random Variable ◽  
Random Series ◽  
Bernoulli Convolution ◽  
Bernoulli Convolutions ◽  
Lévy’S Theorem ◽  
Distribution Matrix ◽  
Necessary And Sufficient

The paper deals with the problem of deepening the Jessen-Wintner theorem for generalized Bernoulli convolutions of a special kind. The main attention is paid to the case when the terms of a random series acquire three values: 0, 1, 2. In the case when the probability that the term of a random series becomes 2 is 0, the corresponding generalized Bernoulli convolutions coincide with classic Bernoulli convolutions, which were actively studied domestic scientists (Pratsovyty M., Turbin G., Torbin G., Honcharenko Ya., Baranovsky O., Savchenko I. and others) as well as foreign researchers (Erdos P., Peres Y., Schlag W, Solomyak B., Albeverio S. and others). The problem of deepening the Jessen-Wintner theorem concerning the necessary and sufficient conditions for the distribution of a probably convergent random series with discrete additions to each of the three pure types, is extremely difficult to formulate and is not completely solved even for classical Bernoulli convolutions. The results of the study are a deepening in relation to the analysis of the Lebesgue structure of random series formed by s-expansions of real numbers. In the case when the corresponding Bernoulli convolution is generated by the sequence 3-n, we have a random variable with independent triple digits, which was studied by scientists in different directions: Lebesgue structure (Chaterji S., Marsaglia G.), topological-metric structure of the distribution spectrum (Pratsovityi M., Turbin G.), fractal analysis of the distribution carrier (Pratsovyty M., Torbin G.), asymptotic properties of the characteristic function at infinity (Honcharenko Ya., Pratsovyty M., Torbin G.). The paper presents certain sufficient conditions for the absolute continuity and singularity of the distribution, with certain restrictions on the stochastic distribution matrix and the asymptotics of the values of the random terms of the series. In the case when the Lebesgue measure of the set of realizations of the generalized Bernoulli convolution is different from zero, it is possible together with Levy's theorem to formulate criteria for belonging of the Bernoulli convolution distribution to each of the three pure Lebesgue types, namely: purely discrete, purely continuous or purely singular.

scholarly journals A lower bound for Garsia’s entropy for certain Bernoulli convolutions

2010 ◽  
Vol 13 ◽  
pp. 130-143 ◽  
Author(s):  
Kevin G. Hare ◽  
Nikita Sidorov
Keyword(s):  
Lower Bound ◽  
High Precision ◽  
Pisot Number ◽  
Bernoulli Convolution ◽  
Pisot Numbers ◽  
Bernoulli Convolutions

AbstractLetβ(1,2) be a Pisot number and letHβdenote Garsia’s entropy for the Bernoulli convolution associated withβ. Garsia, in 1963, showed thatHβ<1 for any Pisotβ. For the Pisot numbers which satisfyxm=xm−1+xm−2++x+1 (withm≥2), Garsia’s entropy has been evaluated with high precision by Alexander and Zagier form=2 and later by Grabner, Kirschenhofer and Tichy form≥3, and it proves to be close to 1. No other numerical values forHβare known. In the present paper we show thatHβ>0.81 for all Pisotβ, and improve this lower bound for certain ranges ofβ. Our method is computational in nature.

Lebesgue structure of asymmetric Bernoulli convolution based on Jacobsthal–Lucas sequence

2020 ◽  
Vol 28 (2) ◽  
pp. 123-130
Author(s):  
Mykola Pratsiovytyi ◽  
Oleg Makarchuk ◽  
Dmytro Karvatsky
Keyword(s):  
Lucas Numbers ◽  
Bernoulli Convolution ◽  
Lucas Sequence ◽  
Bernoulli Convolutions

AbstractWe study the problem of deepening the Jessen–Wintner theorem for asymmetric Bernoulli convolutions. In particular, we investigate the Lebesgue structure of a random incomplete sum of series, whose terms are reciprocal to Jacobsthal–Lucas numbers.

Dimension, measure and infinite Bernoulli convolutions

1998 ◽  
Vol 124 (1) ◽  
pp. 135-149 ◽  
Author(s):  
MICHAEL J. P. COOPER
Keyword(s):  
Lebesgue Measure ◽  
Fractional Dimension ◽  
Bernoulli Convolution ◽  
Absolutely Continuous ◽  
Bernoulli Convolutions

For an infinite Bernoulli convolution whose support is a symmetric Cantor-type set, R. Kershner and A. Wintner [14] proved that the convolution is absolutely continuous or singular depending on whether the Lebesgue measure of the support is positive or zero respectively. Their proof employs the same construction which was developed originally by F. Hausdorff [11] in his celebrated paper, ‘Dimension und äusseres Mass’, where he introduced the notion of a fractional dimension. In this article we exploit this observation and prove a result which extends the theorem of Kershner and Wintner by considering Hausdorff instead of Lebesgue measure for the same class of infinite Bernoulli convolutions.

Primitivity in representations of polycyclic groups

1980 ◽  
Vol 88 (1) ◽  
pp. 15-31 ◽  
Author(s):  
D. L. Harper
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Nilpotent Group ◽  
Infinite Dimension ◽  
Finitely Generated ◽  
Polycyclic Groups

In an earlier paper (5) we showed that a finitely generated nilpotent group which is not abelian-by-finite has a primitive irreducible representation of infinite dimension over any non-absolute field. Here we are concerned primarily with the converse question: Suppose that G is a polycyclic-by-finite group with such a representation, then what can be said about G?

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

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