arXiv subjects

Amlan Banaji

Publications and source records attributed to Amlan Banaji.

17 recordsLinked to original sources

arXiv2026

Fourier transform of nonlinear images of self-similar measures: quantitative aspects

This paper relates to the Fourier decay properties of images of self-similar measures $μ$ on $\mathbb{R}^k$ under nonlinear smooth maps $f \colon \mathbb{R}^k \to \mathbb{R}$. For example, we prove that if the linear parts of the similarities defining $μ$ commute and the graph of $f$ has nonvanishing Gaussian curvature, then the Fourier dimension of the image measure is at least $\max\left\{ \frac{2(2κ_2 - k)}{4 + 2κ_* - k} , 0 \right\}$, where $κ_2$ is the lower correlation dimension of $μ$ and $κ_*$ is the Assouad dimension of the support of $μ$. Under some additional assumptions on $μ$, we use recent breakthroughs in the fractal uncertainty principle to obtain further improvements for the decay exponents. We give several applications to nonlinear arithmetic of self-similar sets $F$ in the line. For example, we prove that if $\dim_{\mathrm H} F > (\sqrt{65} - 5)/4 = 0.765\dots$ then the arithmetic product set $F \cdot F = \{ xy : x,y \in F \}$ has positive Lebesgue measure, while if $\dim_{\mathrm H} F > (-3 + \sqrt{41})/4 = 0.850\dots$ then $F \cdot F \cdot F$ has non-empty interior. One feature of the above results is that they do not require any separation conditions on the self-similar sets.

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arXiv2026

Fourier transform of nonlinear images of self-similar measures: qualitative aspects

The goal of this paper is to establish polynomial Fourier decay for images of self-similar measures $μ$ on $\mathbb{R}^k$ under sufficiently nonlinear real-analytic maps $f \colon \mathbb{R}^k \to \mathbb{R}^d$. For example, we prove that if $f$ is analytic on $\mathbb{R}^k$, its graph does not lie in an affine hyperplane in $\mathbb{R}^{k+d}$, and $μ$ is not supported in an affine hyperplane in $\mathbb{R}^k$, then the image measure has polynomial Fourier decay. Key steps in the proof include establishing a uniform Lojasiewicz-type inequality for self-similar measures, and using the decay of the Fourier transform of $μ$ outside a very small exceptional set of frequencies. As an application of our results, we prove polynomial Fourier decay for self-conformal measures on $\mathbb{C}$ for a large class of complex analytic IFSs which are not self-similar but are conjugate to a linear IFS via an analytic map.

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arXiv2026

Attainable forms of lower spectra

Let $d\in\mathbb{N}$ and $φ\colon(0,1)\to[0,d]$. We prove there exists a set $F\subset\mathbb{R}^d$ whose lower spectrum $\operatorname{dim}^θ_{\mathrm{L}} F$ satisfies $(1-θ)\operatorname{dim}^θ_{\mathrm{L}} F = φ(θ)$ for all $θ\in(0,1)$ if and only if for all $λ,θ\in(0,1)$, \begin{equation*} φ(θ) \leq φ(λθ) - θφ(λ) \leq (1-θ) d. \end{equation*} We also obtain a similar classification result for $\underline{\operatorname{dim}}^θ_{\mathrm{L}} F$. In contrast to the case for Assouad spectra, it is insufficient to consider homogeneous (or uniform) sets. Instead, we follow the approach introduced by Orgoványi--Rutar in arXiv:2510.07013 and proceed via a more general classification result for appropriate two-scale branching functions.

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arXiv2026

Self-similar and self-conformal measures with slow Fourier decay

Given any function $ϕ\colon [0,\infty)\to (0,1]$ satisfying $\lim_{ξ\to\infty}ϕ(ξ) = 0$, we prove the existence of i) self-similar measures and ii) nonlinear $C^{\infty}$ self-conformal measures which are Rajchman and whose Fourier transform $\widehatμ$ satisfies \[ \limsup_{ξ\to\infty}\frac{|\widehatμ(ξ)|}{ϕ(ξ)}>0.\] Moreover, we derive new sufficient conditions for a self-conformal measure to be Rajchman, and construct an explicit self-similar measure $μ$ such that $μ$ almost every $x$ is normal in base $10$ but the sequence $(10^{n}x \mod 1)_{n=1}^{\infty}$ equidistributes extremely slowly.

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arXiv2025

Assouad spectrum of Gatzouras-Lalley carpets

We study the fine local scaling properties of a class of self-affine fractal sets called Gatzouras-Lalley carpets. More precisely, we establish a formula for the Assouad spectrum of all Gatzouras-Lalley carpets as the concave conjugate of an explicit piecewise-analytic function combined with a simple parameter change. Our formula implies a number of novel properties for the Assouad spectrum not previously observed for dynamically invariant sets; in particular, the Assouad spectrum can be a non-trivial differentiable function on the entire domain $(0,1)$ and can be strictly concave on open intervals. Our proof introduces a general framework for covering arguments using techniques developed in the context of multifractal analysis, including the method of types from large deviations theory and Lagrange duality from optimisation theory.

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arXiv2025

Distinct dimensions for attractors of bi-Lipschitz iterated function systems

In this paper, we construct an iterated function system on the line consisting of two bi-Lipschitz contractions whose attractor has distinct lower, Hausdorff, lower box, upper box, and Assouad dimensions, thereby providing negative answers to certain folklore questions. Furthermore, as a by-product of our study of bi-Lipschitz IFSs, we construct IFSs within this family that exhibit interesting fractal behaviour. In particular, we prove the following two statements: (i) There exists a bi-Lipschitz IFS for which the pushforward of any ergodic measure with positive entropy is not exact dimensional; (ii) There exists a bi-Lipschitz IFS whose attractor has empty interior yet positive Lebesgue measure.

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arXiv2025

Interpolating with generalized Assouad dimensions

The $ϕ$-Assouad dimensions are a family of dimensions which interpolate between the upper box and Assouad dimensions. They are a generalization of the well-studied Assouad spectrum with a more general form of scale sensitivity that is often closely related to "phase-transition" phenomena in sets. In this article we establish a number of key properties of the $ϕ$-Assouad dimensions which help to clarify their behaviour. We prove for any bounded doubling metric space $F$ and $α\in\mathbb{R}$ satisfying $\overline{\operatorname{dim}}_{\mathrm{B}}F<α\leq\operatorname{dim}_{\mathrm{A}} F$ that there is a function $ϕ$ so that the $ϕ$-Assouad dimension of $F$ is equal to $α$. We further show that the "upper" variant of the dimension is fully determined by the $ϕ$-Assouad dimension, and that homogeneous Moran sets are in a certain sense generic for these dimensions. Further, we study explicit examples of sets where the Assouad spectrum does not reach the Assouad dimension. We prove a precise formula for the $ϕ$-Assouad dimensions for Galton--Watson trees that correspond to a general class of stochastically self-similar sets, including Mandelbrot percolation. This result follows from two results which may be of general interest: a sharp large deviations theorem for Galton--Watson processes with bounded offspring distribution, and a Borel--Cantelli-type lemma for infinite structures in random trees. Finally, we obtain results on the $ϕ$-Assouad dimensions of overlapping self-similar sets and decreasing sequences with decreasing gaps.

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arXiv2025

Polynomial Fourier decay for fractal measures and their pushforwards

We prove that the pushforwards of a very general class of fractal measures $μ$ on $\mathbb{R}^d$ under a large family of non-linear maps $F \colon \mathbb{R}^d \to \mathbb{R}$ exhibit polynomial Fourier decay: there exist $C,η>0$ such that $|\widehat{Fμ}(ξ)|\leq C|ξ|^{-η}$ for all $ξ\neq 0$. Using this, we prove that if $Φ= \{ φ_a \colon [0,1] \to [0,1] \}_{a \in \mathcal{A}}$ is an iterated function system consisting of analytic contractions, and there exists $a \in \mathcal{A}$ such that $φ_a$ is not an affine map, then every non-atomic self-conformal measure for $Φ$ has polynomial Fourier decay; this result was obtained simultaneously by Algom, Rodriguez Hertz, and Wang. We prove applications related to the Fourier uniqueness problem, Fractal Uncertainty Principles, Fourier restriction estimates, and quantitative equidistribution properties of numbers in fractal sets.

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arXiv2024

Lower box dimension of infinitely generated self-conformal sets

Let $Λ$ be the limit set of an infinite conformal iterated function system and let $F$ denote the set of fixed points of the maps. We prove that the box dimension of $Λ$ exists if and only if \[ \overline{\dim}_{\mathrm B} F\leq \max \{\dim_{\mathrm H} Λ, \underline{\dim}_{\mathrm B} F\}. \] In particular, this provides the first examples of sets of continued fraction expansions with restricted digits for which the box dimension does not exist. More generally, we establish an explicit asymptotic formula for the covering numbers $N_r(Λ)$ in terms of $\dim_{\mathrm H}Λ$ and the covering function $r\mapsto N_r(F)$, where $N_r(\cdot)$ denotes the least number of open balls of radius $r$ required to cover a given set. Such finer scaling information is necessary: in general, the lower box dimension $\underline{\dim}_{\mathrm B} Λ$ is not a function of the Hausdorff dimension of $Λ$ and the upper and lower box dimensions of $F$, and we prove sharp bounds for $\underline{\dim}_{\mathrm B} Λ$ in terms of these three quantities.

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arXiv2024

Attainable forms of intermediate dimensions

The intermediate dimensions are a family of dimensions which interpolate between the Hausdorff and box dimensions of sets. We prove a necessary and sufficient condition for a given function $h(θ)$ to be realized as the intermediate dimensions of a bounded subset of $\mathbb{R}^d$. This condition is a straightforward constraint on the Dini derivatives of $h(θ)$, which we prove is sharp using a homogeneous Moran set construction.

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arXiv2024

Interpolating between Hausdorff and box dimension

Hausdorff and box dimension are two familiar notions of fractal dimension. Box dimension can be larger than Hausdorff dimension, because in the definition of box dimension, all sets in the cover have the same diameter, but for Hausdorff dimension there is no such restriction. This thesis focuses on a family of dimensions parameterised by $θ\in (0,1)$, called the intermediate dimensions, which are defined by requiring that $\mbox{diam}(U) \leq (\mbox{diam}(V))^θ$ for all sets $U,V$ in the cover. We begin by generalising the intermediate dimensions to allow for greater refinement in how the relative sizes of the covering sets are restricted. These new dimensions can recover the interpolation between Hausdorff and box dimension for compact sets whose intermediate dimensions do not tend to the Hausdorff dimension as $θ\to 0$. We also use a Moran set construction to prove a necessary and sufficient condition, in terms of Dini derivatives, for a given function to be realised as the intermediate dimensions of a set. We proceed to prove that the intermediate dimensions of limit sets of infinite conformal iterated function systems are given by the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This applies to sets defined using continued fraction expansions, and has applications to dimensions of projections, fractional Brownian images, and general Hölder images. Finally, we determine a formula for the intermediate dimensions of all self-affine Bedford-McMullen carpets. The functions display features not witnessed in previous examples, such as having countably many phase transitions. We deduce that two carpets have equal intermediate dimensions if and only if the multifractal spectra of the corresponding uniform Bernoulli measures coincide.

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arXiv2024

Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence

Intermediate dimensions were recently introduced to provide a spectrum of dimensions interpolating between Hausdorff and box-counting dimensions for fractals where these differ. In particular, the self-affine Bedford-McMullen carpets are a natural case for investigation, but until now only very rough bounds for their intermediate dimensions have been found. In this paper, we determine a precise formula for the intermediate dimensions $\dim_{\, θ}Λ$ of any Bedford-McMullen carpet $Λ$ for the whole spectrum of $θ\in [0,1]$, in terms of a certain large deviations rate function. The intermediate dimensions exist and are strictly increasing in $θ$, and the function $θ\mapsto \dim_{\, θ}Λ$ exhibits interesting features not witnessed on any previous example, such as having countably many phase transitions, between which it is analytic and strictly concave. We make an unexpected connection to multifractal analysis by showing that two carpets with non-uniform vertical fibres have equal intermediate dimensions if and only if the Hausdorff multifractal spectra of the uniform Bernoulli measures on the two carpets are equal. Since intermediate dimensions are bi-Lipschitz invariant, this shows that the equality of these multifractal spectra is a necessary condition for two such carpets to be Lipschitz equivalent.

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arXiv2024

Assouad type dimensions of infinitely generated self-conformal sets

We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of conformal contractions. Our focus is on the Assouad type dimensions, which give information about the local structure of sets. Under natural separation conditions, we prove a formula for the Assouad dimension and prove sharp bounds for the Assouad spectrum in terms of the Hausdorff dimension of the limit set and dimensions of the set of fixed points of the contractions. The Assouad spectra of the family of examples which we use to show that the bounds are sharp display interesting behaviour, such as having two phase transitions. Our results apply in particular to sets of real or complex numbers which have continued fraction expansions with restricted entries, and to certain parabolic attractors.

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arXiv2024

Intermediate dimensions of infinitely generated attractors

We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of contractions. Our primary focus is on the intermediate dimensions: a family of dimensions depending on a parameter $θ\in [0,1]$ which interpolate between the Hausdorff and box dimensions. Our main results are in the case when all the contractions are conformal. Under a natural separation condition we prove that the intermediate dimensions of the limit set are the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This builds on work of Mauldin and Urbański concerning the Hausdorff and upper box dimension. We give several (often counter-intuitive) applications of our work to dimensions of projections, fractional Brownian images, and general Hölder images. These applications apply to well-studied examples such as sets of numbers which have real or complex continued fraction expansions with restricted entries. We also obtain several results without assuming conformality or any separation conditions. We prove general upper bounds for the Hausdorff, box and intermediate dimensions of infinitely generated attractors in terms of a topological pressure function. We also show that the limit set of a 'generic' infinite iterated function system has box and intermediate dimensions equal to the ambient spatial dimension, where 'generic' can mean either 'full measure' or 'comeagre.'

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arXiv2023

Generalised intermediate dimensions

We introduce a family of dimensions, which we call the $Φ$-intermediate dimensions, that lie between the Hausdorff and box dimensions and generalise the intermediate dimensions introduced by Falconer, Fraser and Kempton. This is done by restricting the relative sizes of the covering sets in a way that allows for greater refinement than in the definition of the intermediate dimensions. We also extend the theory from Euclidean space to a wider class of metric spaces. We prove that these dimensions can be used to 'recover the interpolation' between the Hausdorff and box dimensions of compact subsets for which the intermediate dimensions are discontinuous at $θ=0$, thus providing finer geometric information about such sets. We prove continuity-like results involving the Assouad and lower dimensions, which give a sharp general lower bound for the intermediate dimensions that is positive for all $θ\in (0,1]$ for sets with positive box dimension. We also prove Hölder distortion estimates, a mass distribution principle, and a Frostman type lemma, which we use to study dimensions of product sets.

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arXiv2023

Dimensions of popcorn-like pyramid sets

This article concerns the dimension theory of the graphs of a family of functions which include the well-known 'popcorn function' and its pyramid-like higher-dimensional analogues. We calculate the box and Assouad dimensions of these graphs, as well as the intermediate dimensions, which are a family of dimensions interpolating between Hausdorff and box dimension. As tools in the proofs, we use the Chung$\unicode{x2013}$Erdős inequality from probability theory, higher-dimensional Duffin$\unicode{x2013}$Schaeffer type estimates from Diophantine approximation, and a bound for Euler's totient function. As applications we obtain bounds on the box dimension of fractional Brownian images of the graphs, and on the Hölder distortion between different graphs.

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arXiv2022

Metric Spaces where Geodesics are Never Unique

This article concerns a class of metric spaces, which we call multigeodesic spaces, where between any two distinct points there exist multiple distinct minimising geodesics. We provide a simple characterisation of multigeodesic normed spaces and deduce that $(C([0,1]),||\cdot||_1)$ is an example of such a space, but that finite-dimensional normed spaces are not. We also investigate what additional features are possible in arbitrary metric spaces which are multigeodesic.

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