# 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}
}
• Published 2021
• Mathematics
• Journal of the European Mathematical Society
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 $F… Expand Fourier transform of self-affine measures • Mathematics • 2019 Suppose$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 \inExpand
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#### References

SHOWING 1-10 OF 61 REFERENCES
Fourier transform of self-affine measures
• Mathematics
• 2019
Suppose $F$ is a self-affine set on $\mathbb{R}^d$, $d\geq 2$, which is not a singleton, associated to affine contractions $f_j = A_j + b_j$, $A_j \in \mathrm{GL}(d,\mathbb{R})$, $b_j \inExpand On self-similar sets with overlaps and inverse theorems for entropy in$\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 followingExpand Equidistribution from fractal measures • Mathematics • 2015 We give a fractal-geometric condition for a measure on $$[0,1]$$[0,1] to be supported on points $$x$$x that are normal in base $$n$$n, i.e. such that $$\{n^kx\}_{k\in \mathbb {N}}$${nkx}k∈NExpand Decrease of Fourier coefficients of stationary measures Let $$\mu$$μ be a Borel probability measure on $${\mathrm {SL}}_2(\mathbb {R})$$SL2(R) with a finite exponential moment, and assume that the subgroup $$\varGamma _{\mu }$$Γμ generated by the supportExpand Fourier transforms of Gibbs measures for the Gauss map • Mathematics • 2013 We investigate under which conditions a given invariant measure $$\mu$$μ for the dynamical system defined by the Gauss map $$x \mapsto 1/x \,\,{\mathrm {mod}}\,1$$x↦1/xmod1 is a Rajchman measureExpand On the dimension of Bernoulli convolutions • Mathematics • The Annals of Probability • 2019 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 independentExpand Spectra of Bernoulli convolutions as multipliers in Lp on the circle • Mathematics • 2002 It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution μθ parameterized by a Pisot number θ is countable. Combined with results of R. Salem and P. Sarnak, thisExpand Hausdorff dimension of planar self-affine sets and measures • Mathematics • Inventiones mathematicae • 2019 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 andExpand Fourier dimension and spectral gaps for hyperbolic surfaces • Mathematics • 2017 We obtain an essential spectral gap for a convex co-compact hyperbolic surface $${M=\Gamma\backslash\mathbb H^2}$$M=Γ\H2 which depends only on the dimension $${\delta}$$δ of the limit set. MoreExpand 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 theExpand