Fourier Dimension Research Articles

Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

Given a measure ν \nu on a regular planar domain D D , the Gaussian multiplicative chaos measure of ν \nu studied in this paper is the random measure ν ~ {\widetilde \nu } obtained as the limit of the exponential of the γ \gamma -parameter circle averages of the Gaussian free field on D D weighted by ν \nu . We investigate the dimensional and geometric properties of these random measures. We first show that if ν \nu is a finite Borel measure on D D with exact dimension α > 0 \alpha >0 , then the associated GMC measure ν ~ {\widetilde \nu } is nondegenerate and is almost surely exact dimensional with dimension α − γ 2 2 \alpha -\frac {\gamma ^2}{2} , provided γ 2 2 > α \frac {\gamma ^2}{2}>\alpha . We then show that if ν t \nu _t is a Hölder-continuously parameterized family of measures, then the total mass of ν ~ t {\widetilde \nu }_t varies Hölder-continuously with t t , provided that γ \gamma is sufficiently small. As an application we show that if γ > 0.28 \gamma >0.28 , then, almost surely, the orthogonal projections of the γ \gamma -Liouville quantum gravity measure μ ~ {\widetilde \mu } on a rotund convex domain D D in all directions are simultaneously absolutely continuous with respect to Lebesgue measure with Hölder continuous densities. Furthermore, μ ~ {\widetilde \mu } has positive Fourier dimension almost surely.

Explicit Salem sets and applications to metrical Diophantine approximation

Let $Q$ be an infinite subset of $\mathbb{Z}$, let $\Psi: \mathbb{Z} \rightarrow [0,\infty)$ be positive on $Q$, and let $\theta \in \mathbb{R}$. Define $$ E(Q,\Psi,\theta) = \{ x \in \mathbb{R} : \| q x - \theta \| \leq \Psi(q) \text{ for infinitely many $q \in Q$} \}. $$ We prove a lower bound on the Fourier dimension of $E(Q,\Psi,\theta)$. This generalizes theorems of Kaufman and Bluhm and yields new explicit examples of Salem sets. We give applications to metrical Diophantine approximation, including determining the Hausdorff dimension of $E(Q,\Psi,\theta)$ in new cases. We also prove a higher-dimensional analog of our result.

On Assouad Dimension and Arithmetic Progressions in Sets Defined by Digit Restrictions

We show that the set defined by digit restrictions contains arbitrarily long arithmetic progressions if and only if its Assouad dimension is one. Moreover, we show that for any $$0\le s\le 1$$ , there exists some set on $$\mathbb {R}$$ with Hausdorff dimension s whose Fourier dimension is zero and it contains arbitrarily long arithmetic progressions.

On the Fourier analytic structure of the Brownian graph

In a previous article (\textit{Int. Math. Res. Not.} 2014, 2730--2745) T. Orponen and the authors proved that the Fourier dimension of the graph of any real-valued function on $\mathbb{R}$ is bounded above by $1$. This partially answered a question of Kahane ('93) by showing that the graph of the Wiener process $W_t$ (Brownian motion) is almost surely not a Salem set. In this article we complement this result by showing that the Fourier dimension of the graph of $W_t$ is almost surely $1$. In the proof we introduce a method based on Ito calculus to estimate Fourier transforms by reformulating the question in the language of Ito drift-diffusion processes and combine it with the classical work of Kahane on Brownian images.

Fourier dimension and spectral gaps for hyperbolic surfaces

We obtain an essential spectral gap for a convex co-compact hyperbolic surface $M=\Gamma\backslash\mathbb H^2$ which depends only on the dimension $\delta$ of the limit set. More precisely, we show that when $\delta>0$ there exists $\varepsilon_0=\varepsilon_0(\delta)>0$ such that the Selberg zeta function has only finitely many zeroes $s$ with $\Re s>\delta-\varepsilon_0$. The proof uses the fractal uncertainty principle approach developed by Dyatlov-Zahl [arXiv:1504.06589]. The key new component is a Fourier decay bound for the Patterson-Sullivan measure, which may be of independent interest. This bound uses the fact that transformations in the group $\Gamma$ are nonlinear, together with estimates on exponential sums due to Bourgain which follow from the discretized sum-product theorem in $\mathbb R$.

Perfect fractal sets with zero Fourier dimension and arbitrarily long arithmetic progressions

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.

Fractally Fourier decimated homogeneous turbulent shear flow in noninteger dimensions.

Time evolution of the fully resolved incompressible homogeneous turbulent shear flow in noninteger Fourier dimensions is numerically investigated. The Fourier dimension of the flow field is extended from the integer value 3 to the noninteger values by projecting the Navier-Stokes equation on the fractal set of the active Fourier modes with dimensions 2.7≤d≤3.0. The results of this study revealed that the dynamics of both large and small scale structures are nontrivially influenced by changing the Fourier dimension d. While both turbulent production and dissipation are significantly hampered as d decreases, the evolution of their ratio is almost independent of the Fourier dimension. The mechanism of the energy distribution among different spatial directions is also impeded by decreasing d. Due to this deficient energy distribution, turbulent field shows a higher level of the large-scale anisotropy in lower Fourier dimensions. In addition, the persistence of the vortex stretching mechanism and the forward spectral energy transfer, which are three-dimensional turbulence characteristics, are examined at changing d, from the standard case d=3.0 to the strongly decimated flow field for d=2.7. As the Fourier dimension decreases, these forward energy transfer mechanisms are strongly suppressed, which in turn reduces both the small-scale intermittency and the deviation from Gaussianity. Besides the energy exchange intensity, the variations of d considerably modify the relative weights of local to nonlocal triadic interactions. It is found that the contribution of the nonlocal triads to the total turbulent kinetic energy exchange increases as the Fourier dimension increases.

Lumley decomposition of turbulent boundary layer at high Reynolds numbers

The decomposition proposed by Lumley in 1966 is applied to a high Reynolds number turbulent boundary layer. The experimental database was created by a hot-wire rake of 143 probes in the Laboratoire de Mécanique de Lille wind tunnel. The Reynolds numbers based on momentum thickness (Reθ) are 9800 and 19 100. Three-dimensional decomposition is performed, namely, proper orthogonal decomposition (POD) in the inhomogeneous and bounded wall-normal direction, Fourier decomposition in the homogeneous spanwise direction, and Fourier decomposition in time. The first POD modes in both cases carry nearly 50% of turbulence kinetic energy when the energy is integrated over Fourier dimensions. The eigenspectra always peak near zero frequency and most of the large scale, energy carrying features are found at the low end of the spectra. The spanwise Fourier mode which has the largest amount of energy is the first spanwise mode and its symmetrical pair. Pre-multiplied eigenspectra have only one distinct peak and it matches the secondary peak observed in the log-layer of pre-multiplied velocity spectra. Energy carrying modes obtained from the POD scale with outer scaling parameters. Full or partial reconstruction of turbulent velocity signal based only on energetic modes or non-energetic modes revealed the behaviour of urms in distinct regions across the boundary layer. When urms is based on energetic reconstruction, there exists (a) an exponential decay from near wall to log-layer, (b) a constant layer through the log-layer, and (c) another exponential decay in the outer region. The non-energetic reconstruction reveals that urms has (a) an exponential decay from the near-wall to the end of log-layer and (b) a constant layer in the outer region. Scaling of urms using the outer parameters is best when both energetic and non-energetic profiles are combined.

On polynomial configurations in fractal sets

We show that subsets of $\mathbb{R}^n$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form \begin{align*} ( x ,\, x + A_1 y ,\, \dots,\, x + A_{k-1} y ,\, x + A_k y + Q(y) e_n ), \quad x \in \mathbb{R}^n,\ y \in \mathbb{R}^m, \end{align*} where $A_i$ are real $n \times m$ matrices, $Q$ is a real polynomial in $m$ variables and $e_n = (0,\dots,0,1)$.

Fourier dimension of random images

Given a compact set of real numbers, a random Cm+α-diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number s, almost surely has Fourier dimension greater than or equal to s/(m+α). This is used to show that every Borel subset of the real numbers of Hausdorff dimension s is Cm+α-equivalent to a set of Fourier dimension greater than or equal to s/(m+α). In particular every Borel set is diffeomorphic to a Salem set, and the Fourier dimension is not invariant under Cm-diffeomorphisms for any m.

On the Fourier dimension and a modification

We give a sufficient condition for the Fourier dimension of a countable union of sets to equal the supremum of the Fourier dimensions of the sets in the union, and show by example that the Fourier dimension is not countably stable in general. A natural approach to finite stability of the Fourier dimension for sets would be to try to prove that the Fourier dimension for measures is finitely stable, but we give an example showing that it is not in general. We also describe some situations where the Fourier dimension for measures is stable or is stable for all but one value of some parameter. Finally we propose a way of modifying the definition of the Fourier dimension so that it becomes countably stable, and show that for each s there is a class of sets such that a measure has modied Fourier dimension greater than or equal to s if and only if it annihilates all sets in the class.

The Fourier dimension of Brownian limsup fractals

Robert Kaufman’s proof that the set of rapid points of Brownian motion has a Fourier dimension equal to its Hausdorff dimension was first published in 1974. Some of the original arguments presented by Kaufman are not clear, hence this paper presents a new version of the construction and incorporates some recent results in order to establish a slightly weaker version of Kaufman’s theorem. This weaker version still implies that the Fourier and Hausdorff dimensions of Brownian rapid points are equal. The method of proof can then be extended to show that functionally determined rapid points of Brownian motion also form Salem sets for absolutely continuous functions of finite energy.

Sets of Salem type and sharpness of the 𝐿²-Fourier restriction theorem

We construct Salem sets on the real line with endpoint Fourier decay and near-endpoint regularity properties. This complements a result of Łaba and Pramanik, who obtained near-endpoint Fourier decay and endpoint regularity properties. We then modify the construction to extend a theorem of Hambrook and Łaba to show sharpness of the L 2 L^2 -Fourier restriction estimate by Mockenhaupt and Bak-Seeger, including the case where the Hausdorff and Fourier dimension do not coincide.

The Fourier Dimension is Not Finitely Stable

The Fourier dimension is not, in general, stable under finite unions of sets. Moreover, the stability of the Fourier dimension on particular pairs of sets is independent from the stability of the compact Fourier dimension.

The progressive development of turbulence statistics and its impact on wind power predictability

Wind resource assessment is a critical parameter in a diverse range of considerations within the built environment. Engineers and scientists, engaging in building design, energy conservation/application and air-quality/air-pollution control measures, need to be cognisant of how the associated wind resource imposes increased complexities in their design and modelling processes. In this regard, the morphological heterogeneities within these environments, present significant challenges to quantifying the resource and its turbulent characteristics.This paper presents three aspects of turbulence assessment within the built environment. Firstly, an analysis of how turbulence is currently quantified is considered. The industry standard, TI (Turbulence Intensity) is compared with a proposed alternative metric described as TDf modelling (Turbulent Fourier Dimension). Secondly, the application of the turbulence assessment is considered with respect to how TI affects the productivity of small/micro wind turbines in complex environments though Gaussian distribution analysis. Finally, an extended discussion on current developments such as the concept of a turbulence rose and the ongoing development of statistical modelling is presented.

On Fourier Analytic Properties of Graphs

We study the Fourier dimensions of graphs of real-valued functions defined on the unit interval [0,1]. Our results imply that the graph of fractional Brownian motion is almost surely not a Salem set, answering in part a question of Kahane from 1993, and that the graph of a Baire typical function in C[0,1] has Fourier dimension 0.

Higuchi dimension of digital images.

There exist several methods for calculating the fractal dimension of objects represented as 2D digital images. For example, Box counting, Minkowski dilation or Fourier analysis can be employed. However, there appear to be some limitations. It is not possible to calculate only the fractal dimension of an irregular region of interest in an image or to perform the calculations in a particular direction along a line on an arbitrary angle through the image. The calculations must be made for the whole image. In this paper, a new method to overcome these limitations is proposed. 2D images are appropriately prepared in order to apply 1D signal analyses, originally developed to investigate nonlinear time series. The Higuchi dimension of these 1D signals is calculated using Higuchi's algorithm, and it is shown that both regions of interests and directional dependencies can be evaluated independently of the whole picture. A thorough validation of the proposed technique and a comparison of the new method to the Fourier dimension, a common two dimensional method for digital images, are given. The main result is that Higuchi's algorithm allows a direction dependent as well as direction independent analysis. Actual values for the fractal dimensions are reliable and an effective treatment of regions of interests is possible. Moreover, the proposed method is not restricted to Higuchi's algorithm, as any 1D method of analysis, can be applied.

Extracting the Excitonic Hamiltonian of the Fenna-Matthews-Olson Complex Using Three-Dimensional Third-Order Electronic Spectroscopy

We extend traditional two-dimensional (2D) electronic spectroscopy into a third Fourier dimension without the use of additional optical interactions. By acquiring a set of 2D spectra evenly spaced in waiting time and dividing the area of the spectra into voxels, we can eliminate population dynamics from the data and transform the waiting time dimension into frequency space. The resultant 3D spectrum resolves quantum beating signals arising from excitonic coherences along the waiting frequency dimension, thereby yielding up to 2 n-fold redundancy in the set of frequencies necessary to construct a complete set of n excitonic transition energies. Using this technique, we have obtained, to our knowledge, the first fully experimental set of electronic eigenstates for the Fenna-Matthews-Olson (FMO) antenna complex, which can be used to improve theoretical simulations of energy transfer within this protein. Whereas the strong diagonal peaks in the 2D rephasing spectrum of the FMO complex obscure all but one of the crosspeaks at 77 K, extending into the third dimension resolves 19 individual peaks. Analysis of the independently collected nonrephasing data provides the same information, thereby verifying the calculated excitonic transition energies. These results enable one to calculate the Hamiltonian of the FMO complex in the site basis by fitting to the experimental linear absorption spectrum.

Exceptional sets for self-affine fractals

AbstractUnder certain conditions the ‘singular value function’ formula gives the Hausdorff dimension of self-affine fractals for almost all parameters in a family. We show that the size of the set of exceptional parameters is small both in the sense of Hausdorff dimension and Fourier dimension.

Distributional dimension of fractal sets in local fields

The distributional dimension of fractal sets in ℝn has been systematically studied by Triebel by virtue of the theory of function spaces. In this paper, we first discuss some important properties about the B-type spaces and the F-type spaces on local fields, then we give the definition of the distributional dimension dimD in local fields and study the relations between distributional dimension and Hausdorff dimension. Moreover, the analysis expression of the Hausdorff dimension is given. Lastly, we define the Fourier dimension in local fields, and obtain the relations among all the three dimensions.