# Trigonometric series and self-similar sets

@article{Li2021TrigonometricSA, title={Trigonometric series and self-similar sets}, author={Jialun Li and Tuomas Sahlsten}, journal={Journal of the European Mathematical Society}, year={2021} }

Let $F$ be a self-similar set on $\mathbb{R}$ associated to contractions $f_j(x) = r_j x + b_j$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$, such that $F$ is not a singleton. We prove that if $\log r_i / \log r_j$ is irrational for some $i \neq j$, then $F$ is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of $F$. No separation conditions are assumed on $F$. We establish our result by showing that every self-similar measure $\mu$ on $F… Expand

#### 13 Citations

Fourier transform of self-affine measures

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- 2019

Suppose $F$ is a self-affine set on $\mathbb{R}^d$, $d\geq 2$, which is not a singleton, associated to affine contractions $f_j = A_j + b_j$, $A_j \in \mathrm{GL}(d,\mathbb{R})$, $b_j \in… Expand

On the decay of the Fourier transform of self-conformal measures

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Let $\Phi$ be a $C^{1+\gamma}$ smooth IFS on an interval $J\subset \mathbb{R}$, where $\gamma>0$.
We provide mild conditions on the derivative cocycle that ensure that all non-atomic self conformal… Expand

On normal numbers and self-similar measures

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Let b ≥ 2 be an integer. A real number x is said to be normal in base b if the sequence (bx)n=1 is uniformly distributed modulo one. For a real number x, being normal in base b indicates that the… Expand

Pointwise normality and Fourier decay for self-conformal measures

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- 2020

Let Φ be a C smooth IFS on R, where γ > 0. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points x that are absolutely normal. That… Expand

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- 2021

We discuss the problem of determining the dimension of self-similar sets and measures on R. We focus on the developments of the last four years. At the end of the paper, we survey recent results… Expand

Fourier decay for self-similar measures

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- 2019

We prove that, after removing a zero Hausdorff dimension exceptional set of parameters, all self-similar measures on the line have a power decay of the Fourier transform at infinity. In the… Expand

Fourier decay for homogeneous self-affine measures

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- 2021

lim p μpξq “ 0, as |ξ| Ñ 8, where |ξ| is a norm (say, the Euclidean norm) of ξ P Rd. Whereas absolutely continuous measures are Rajchman by the Riemann-Lebesgue Lemma, it is a subtle question to… Expand

On the Rajchman property for self-similar measures

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- 2019

For classical Bernoulli convolutions, the Rajchman property, i.e. the convergence to zero at infinity of the Fourier transform, was characterized by successive works of Erdos [2] and Salem [12]. We… Expand

Logarithmic Fourier decay for self conformal measures

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We prove that the Fourier transform of a self conformal measure on R decays to 0 at infinity at a logarithmic rate, unless the following holds: The underlying IFS is smoothly conjugated to an IFS… Expand

On the Rajchman property for self-similar measures on $\mathbb{R}^{d}$

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- 2021

We establish a complete algebraic characterization of self-similar iterated function systems Φ on Rd, for which there exists a positive probability vector p so that the Fourier transform of the… Expand

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