Absolute continuity of Bernoulli convolutions for algebraic parameters

@article{Varju2016AbsoluteCO,
  title={Absolute continuity of Bernoulli convolutions for algebraic parameters},
  author={P'eter P'al Varj'u},
  journal={Journal of the American Mathematical Society},
  year={2016}
}
  • P. P. Varj'u
  • Published 31 January 2016
  • Computer Science
  • Journal of the American Mathematical Society
<p>We prove that Bernoulli convolutions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript lamda"> <mml:semantics> <mml:msub> <mml:mi>μ<!-- μ --></mml:mi> <mml:mi>λ<!-- λ --></mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mu _\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are absolutely continuous provided the parameter <inline-formula content-type… 
View PDF on arXiv
38 Citations
Highly Influential Citations
6
Background Citations
23
Methods Citations
3
Results Citations
5

38 Citations

On the dimension of Bernoulli convolutions

The Bernoulli convolution with parameter $\lambda\in(0,1)$ is the probability measure $\mu_\lambda$ that is the law of the random variable $\sum_{n\ge0}\pm\lambda^n$, where the signs are independent…

On the dimension of Bernoulli convolutions for all transcendental parameters

The Bernoulli convolution $\nu_\lambda$ with parameter $\lambda\in(0,1)$ is the probability measure supported on $\mathbf{R}$ that is the law of the random variable $\sum\pm\lambda^n$, where the…

On the Hausdorff Dimension of Bernoulli Convolutions

An expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices is given that shows that one can calculate the Hausdorff dimension of the Bernouelli convolution to arbitrary given accuracy whenever $\beta $ is algebraic.

Typical absolute continuity for classes of dynamically defined measures

On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions

We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli…

On the dimension of Furstenberg measure for $${ SL}_{2}(\mathbb {R})$$SL2(R) random matrix products

Let $$\mu $$μ be a measure on $${ SL}_{2}({\mathbb {R}})$$SL2(R) generating a non-compact and totally irreducible subgroup, and let $$\nu $$ν be the associated stationary (Furstenberg) measure for…

Arbeitsgemeinschaft mit aktuellem Thema : Additive combinatorics , entropy and fractal geometry

This series of twenty-one talks aims to cover many of the recent developments in the dimension theory of self-similar measures and their projections and intersections, especially on Furstenberg's…

On the dimension of Furstenberg measure for random matrix products

Letμ be ameasure on SL2(R) generating a non-compact and totally irreducible subgroup, and let ν be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that…

Hausdorff dimension of planar self-affine sets and measures

Let $$X={\bigcup }{\varphi }_{i}X$$X=⋃φiX be a strongly separated self-affine set in $${\mathbb {R}}^2$$R2 (or one satisfying the strong open set condition). Under mild non-conformality and…

A Dichotomy for the dimension of solenoidal attractors on high dimensional space

We study dynamical systems generated by skew products: T : [0, 1) × C→ [0, 1) × C T (x, y) = (bx mod 1, γy + φ(x)) where integer b ≥ 2, γ ∈ C such that 0 < |γ| < 1, and φ is a real analytic…

References

SHOWING 1-10 OF 34 REFERENCES

On the dimension of Bernoulli convolutions

The Bernoulli convolution with parameter $\lambda\in(0,1)$ is the probability measure $\mu_\lambda$ that is the law of the random variable $\sum_{n\ge0}\pm\lambda^n$, where the signs are independent…

On the Smoothness Properties of a Family of Bernoulli Convolutions

Let L (u, a),-oo < u < + oo denote the Fourier-Stieltjes transform, 00 f eluoda(x), of a distribution function u(x),-oo < x < + co. Thus if 00 /3(x) is the distribution function which is 0, J, 1…

Arithmetic properties of Bernoulli convolutions

Here and in the following we shall suppose that this condition is satisfied. It is easy to show that F(x, r) is continuous. Since it can be shown (see [4]) that it is always "pure," i.e. either…

On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions

We prove that the set of exceptional $${\lambda\in (1/2,1)}$$λ∈(1/2,1) such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli…

On self-similar sets with overlaps and inverse theorems for entropy in $\mathbb{R}^d$

We study self-similar sets and measures on $\mathbb{R}^{d}$. Assuming that the defining iterated function system $\Phi$ does not preserve a proper affine subspace, we show that one of the following…

Distribution functions and the Riemann zeta function

The present paper starts with a systematic study of distribution functions in ^-dimensional space and in particular of their infinite convolutions representing, in the language of the calculus of…

Quantitative Density under Higher Rank Abelian Algebraic Toral Actions

We generalize Bourgain-Lindenstrauss-Michel-Venkatesh's recent one-dimensional quantitative density result to abelian algebraic actions on higher dimensional tori. Up to finite index, the group…

Sumset and Inverse Sumset Inequalities for Differential Entropy and Mutual Information

The results include differential-entropy versions of Ruzsa's triangle inequality, the Plünnecke-Ruzsa inequality, and the Balog-Szemerédi-Gowers lemma, and a differential entropy version of a Freiman-type inverse-sumset theorem, which can be seen as a quantitative converse to the entropy power inequality.

Fractal geometry

Editor's note: The following articles by Steven G. Krantz and Benoit B. Mandelbrot have an unusual history. In the fall of 1988, Krantz asked the Bulletin of the American Mathematical Society Book…

Elements of Information Theory

The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.