Fourier decay of self-similar measures and self-similar sets of uniqueness
Péter P. Varjú, Han Yu
备注：
16 pages, this is the final accepted version to appear in Ann. PDE, revision based on referee's comments, added assumption in Corollary 1.6 that r^-1 is an algebraic integer
In this paper, we investigate the Fourier transform of self-similar measures on R. We provide quantitative decay rates of Fourier transform of some self-similar measures. Our method is based on random walks on lattices and Diophantine approximation in number fields. We also completely identify all self-similar sets which are sets of uniqueness. This generalizes a classical result of Salem and Zygmund.
We obtain new Fourier interpolation and -uniqueness results in all dimensions, extending methods and results by the first author and M. Sousa, and by the second author. We show that the only Schwartz function which, together with its Fourier transform, vanishes on surfaces close to the origin-centered spheres whose radius are square roots of integers, is the zero function. In the radial case, these surfaces are spheres with perturbed radii, while in the non-radial case, they can be graphs of continuous functions over the sphere. As an application, we translate our perturbed Fourier uniqueness …
This paper builds upon two key principles behind the Bourgain-Dyatlov quantitative uniqueness theorem for functions with Fourier transform supported in an Ahlfors regular set. We first provide a characterization of when a quantitative uniqueness theorem holds for functions with very quickly decaying Fourier transform, thereby providing an extension of the classical Paneah-Logvinenko-Sereda theorem. Secondly, we derive a transference result which converts a quantitative uniqueness theorem for functions with fast decaying Fourier transform to one for functions with Fourier transform supported on…
Let $K$ be a totally real number field of degree $n \geq 2$. The inverse different of $K$ gives rise to a lattice in $\mathbb{R}^n$. We prove that the space of Schwartz Fourier eigenfunctions on $\mathbb{R}^n$ which vanish on the "component-wise square root" of this lattice, is infinite dimensional. The Fourier non-uniqueness set thus obtained is a discrete subset of the union of all spheres $\sqrt{m}S^{n-1}$ for integers $m \geq 0$ and, as $m \rightarrow \infty$, there are $\sim c_{K} m^{n-1}$ many points on the $m$-th sphere for some explicit constant $c_{K}$, proportional to the square root…
In this article, we prove that a cone is a Heisenberg uniqueness pair corresponding to sphere as long as the cone does not completely recline on the level surface of any homogeneous harmonic polynomial on $\mathbb R^n.$ We derive that $\left(S^2, \text{ paraboloid}\right)$ and $\left(S^2, \text{ geodesic of } S_r(o)\right)$ are Heisenberg uniqueness pairs for a class of certain symmetric finite Borel measures in $\mathbb R^3.$ Further, we correlate the problem of Heisenberg uniqueness pairs to the sets of injectivity for the spherical mean operator.
We prove, under certain conditions on $(α,β)$, that each Schwartz function $f$ such that $f(\pm n^α) = \hat{f}(\pm n^β) = 0, \forall n \ge 0$ must vanish identically, complementing a series of recent results involving uncertainty principles, such as the pointwise interpolation formulas by Radchenko and Viazovska and the Meyer-Guinnand construction of self-dual crystaline measures.
We obtain a Fourier dimension estimate for sets of exact approximation order introduced by Bugeaud for certain approximation functions $ψ$. This Fourier dimension estimate implies that these sets of exact approximation order contain normal numbers.
It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets $Λ$. We show that the same is true for much wider spaces of continuous functions. In particular, Sobolev spaces have this property whenever $S$ is a set of infinite measure having "periodic gaps". The periodicity condition is crucial. For sets $S$ with randomly distributed gaps, we show that the uniformly discrete sets $Λ$ satisfy a strong non-uniqueness property: Every discrete function $c(λ)\in l^2(Λ)$ can be interpolated by an analytic $L^2$-function with sp…
We exhibit the first explicit examples of Salem sets in $\mathbb{Q}_p$ of every dimension $0 < α< 1$ by showing that certain sets of well-approximable $p$-adic numbers are Salem sets. We construct measures supported on these sets that satisfy essentially optimal Fourier decay and upper regularity conditions, and we observe that these conditions imply that the measures satisfy strong Fourier restriction inequalities. We also partially generalize our results to higher dimensions. Our results extend theorems of Kaufman, Papadimitropoulos, and Hambrook from the real to the $p$-adic setting.
By considering a Moran-type construction of fractals on $[0,1]$, we show that for any $0\le s\le 1$, there exists some Moran fractal set, which is perfect, with Hausdorff dimension $s$ whose Fourier dimension is zero and it contains arbitrarily long arithmetic progressions.
A $(d,k)$-set is a subset of $\mathbb{R}^d$ containing a $k$-dimensional unit ball of all possible orientations. Using an approach of D.~Oberlin we prove various Fourier dimension estimates for compact $(d,k)$-sets. Our main interest is in restricted $(d,k)$-sets, where the set only contains unit balls with a restricted set of possible orientations $Γ$. In this setting our estimates depend on the Hausdorff dimension of $Γ$ and can sometimes be improved if additional geometric properties of $Γ$ are assumed. We are led to consider cones and prove that the cone in $\mathbb{R}^{d+1}$ has Fourier d…
We construct a function that lies in $L^p(\mathbb{R}^d)$ for every $p \in (1,\infty]$ and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kovač's maximal restriction principle to show that the same full-dimensional set is avoided by any Borel measure satisfying a nontrivial Fourier restriction theorem. As a consequence of a near-optimal fractal restriction theorem of Łaba and Wang, we hence prove a lack of valid relations between the Hausdorff dimension of a set and the range of possible Fourier restriction exponents for measures supported i…
We study the Fourier transforms $\widehatμ(ξ)$ of non-atomic Gibbs measures $μ$ for uniformly expanding maps $T$ of bounded distortions on $[0,1]$ or Cantor sets with strong separation. When $T$ is totally non-linear, then $\widehatμ(ξ) \to 0$ at a polynomial rate as $|ξ| \to \infty$.
In this paper we establish Lebesgue-type inequalities for $2π$-periodic functions $f$, which are defined by generalized Poisson integrals of the functions $φ$ from $L_{p}$, $1\leq p< \infty$. In these inequalities uniform norms of deviations of Fourier sums $\| f-S_{n-1} \|_{C}$ are expressed via best approximations $E_{n}(φ)_{L_{p}}$ of functions $φ$ by trigonometric polynomials in the metric of space $L_{p}$. We show that obtained estimates are asymptotically best possible.
On the sets of $2π$-periodic functions $f$, which are defined with a help of $(ψ, β)$-integrals of the functions $φ$ from $L_{1}$, we establish Lebesgue-type inequalities, in which the uniform norms of deviations of Fourier sums are expressed via the best approximations by trigonometric polynomials of the functions $φ$. We prove that obtained estimates are best possible, in the case when the sequences $ψ(k)$ decrease to zero faster than any power function. In some important cases we establish the asymptotic equalities for the exact upper boundaries of uniform approximations by Fourier sums on …
We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in threedimensional euclidean space. Our approach exploits in a crucial way the underlying hyperbolic geometry, which leads to a novel notion of strong transversality and corresponding "exceptional" sets. For the division of these exceptional sets we make crucial and perhaps surprising use of a lemma on level sets for sufficiently smooth one-variate functions from a previous article of ours.
This paper considers the problem of restricting the short-time Fourier transform to domains of nonzero measure in the plane and studies sampling bounds of such systems. In particular, we give a quantitative estimate for the lower sampling bound in the case of Hermite windows and derive a sufficient condition for a large class of windows in terms of a certain planar density. On the way, we prove a Remez-type inequality for polyanalytic functions. MSC2010: 42C40, 46E15, 46E20, 42C15
Let $f:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ be a hyperbolic rational map of degree $d \geq 2$, and let $J \subset \mathbb{C}$ be its Julia set. We prove that $J$ always has positive Fourier dimension. The case where $J$ is included in a circle follows from a recent work of Sahlsten and Stevens, see arXiv:2009.01703. In the case where $J$ is not included in a circle, we prove that a large family of probability measures supported on $J$ exhibit polynomial Fourier decay: our result applies in particular to the measure of maximal entropy and to the conformal measure.
Two measurable sets $S, Λ\subseteq \mathcal{R}^d$ form a Heisenberg uniqueness pair, if every bounded measure $μ$ with support in S whose Fourier transform vanishes on Λ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathcal{R}^d$. As a corollary we obtain a new, surprising version of the classical Cramér-Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of…
