Related papers: Spectral property of Cantor measures with consecut…

For any $\alpha\in(0,d)$, we construct Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $\alpha$ such that the associated natural measure $\mu$ obeys the restriction estimate $\| \widehat{f d\mu} \|_{p} \leq C_p \| f \|_{L^2(\mu)}$ for…

We study maximal averages associated with singular measures on $\rr$. Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension $1 - \epsilon$, $0 \leq \epsilon < {1/3}$ for which the…

In this paper, we add to the characterization of the Fourier spectra for Bernoulli convolution measures. These measures are supported on Cantor subsets of the line. We prove that performing an odd additive translation to half the canonical…

The Doob convergence theorem implies that the set of divergence of any martingale has measure zero. We prove that, conversely, any $G\_{\delta\sigma}$ subset of the Cantor space with Lebesgue-measure zero can be represented as the set of…

We investigate some relations between number theory and spectral measures related to the harmonic analysis of a Cantor set. Specifically, we explore ways to determine when an odd natural number $m$ generates a complete or incomplete Fourier…

It is well known that a purely unrectifiable set cannot support a harmonic measure which is absolutely continuous with respect to the Hausdorff measure of this set. We show that nonetheless there exist elliptic operators on (purely…

Take an interval $[t, t+1]$ on the $x$-axis together with the same interval on the $y$-axis and let $\rho$ be the normalized one-dimensional Lebesgue measure on this set of two segments. Continuing the work done by Lai, Liu and Prince…

In this paper, we consider spectral properties of Riesz product measures supported on homogeneous Cantor sets and we show the existence of spectral measures with arbitrary Hausdorff dimensions, including non-atomic zero-dimensional spectral…

We obtain a complete description for a probability measure to be doubling on an arbitrarily given uniform Cantor set. The question of which doubling measures on such a Cantor set can be extended to a doubling measure on [0; 1] is also…

Fix a sequence of integers $Q=\{q_n\}_{n=1}^\infty$ such that $q_n$ is greater than or equal to 2 for all $n$. In this paper, we improve upon results by J. Galambos and F. Schweiger showing that almost every (in the sense of Lebesgue…

We construct a class of homogeneous Cantor-Moran measures with all contraction ratios being reciprocal of integers, and prove that they are pointwise absolutely normal. Our approach relies on methods developed by Davenport, Erd{\H{o}}s, and…

For a family of fractal measures, we find an explicit Fourier duality. The measures in the pair have compact support in $\br^d$, and they both have the same matrix scaling. But the two use different translation vectors, one by a subset $B$…

We show conditions on $k$ such that any number $x$ in the interval $[0, k/2]$ can be represented in the form $x_1^{a_1} x_2^{a_2} + x_3^{a_3} x_4^{a_4} + \cdots + x_{k-1}^{a_{k-1}} x_k^{a_k}$, where the exponents $a_{2i-1}$ and $a_{2i}$ are…

A bounded measurable set $\Omega$, of Lebesgue measure 1, in the real line is called spectral if there is a set $\Lambda$ of real numbers ("frequencies") such that the exponential functions $e_\lambda(x) = \exp(2\pi i \lambda x)$,…

In this work we investigate the question of constructions of the possible Fourier bases $E(\Lambda)=\{e^{2\pi i \lambda x}:\lambda\in\Lambda\}$ for the Hilbert space $L^2(\mu_4)$, where $\mu_4$ is the standard middle-fourth Cantor measure…

Let $S \subseteq \mathbb{N}$ have the property that for each $k \in S$ the set $(S - k) \cap \mathbb{N} \setminus S$ has asymptotic density $0$. We prove that there exists a basic sequence $Q$ where the set of numbers $Q$-normal of all…

Motivated by Erd\H{o}s' ternary conjecture and by recent work of Cui--Ma--Jiang [``Geometric progressions meet Cantor sets'', \textit{Chaos Solitons Fractals} \textbf{163} (2022), 112567.] on intersections between geometric progressions and…

We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension. Let a be any real number greater than or equal to 2 and let b be any non-negative real less than or equal…

We study spectral measures generated by infinite convolution products of discrete measures generated by Hadamard triples, and we present sufficient conditions for the measures to be spectral, generalizing a criterion by Strichartz. We then…

Let $\mu$ be a Borel probability measure with compact support. We consider exponential type orthonormal bases, Riesz bases and frames in $L^2(\mu)$. We show that if $L^2(\mu)$ admits an exponential frame, then $\mu$ must be of pure type. We…