arXiv subjects

Jonathan M. Fraser

Publications and source records attributed to Jonathan M. Fraser.

At least 19 recordsLinked to original sources

arXiv2026

On the Marstrand projection theorem for the Assouad spectrum

Marstrand's projection theorem states that the Hausdorff dimension of the orthogonal projection of a Borel set in the plane onto lines is constant almost surely. This property extends to other notions of dimension, such as box and packing dimensions, but does not hold for the Assouad dimension. In this paper, we show that Marstrand's projection theorem also fails for the quasi-Assouad dimension and the Assouad spectrum, which interpolates between the upper box and quasi-Assouad dimensions. Additionally, we establish an almost sure lower bound for the Assouad spectrum of the projections using capacity-theoretic dimension profiles, and an almost sure upper bound for projections of bounded planar sets via an incidence geometry-inspired tube-counting argument. As an application, for a parametrised family of homogeneous self-similar sets, we obtain an almost sure upper bound for the Assouad spectrum which beats the trivial upper bound coming from the upper box dimension.

math.MG↗

arXiv2026

Quantitative flatness and obstructions in Fourier analysis

Three important problems in Fourier analysis are the Fourier restriction problem, the $L^p$-improving problem, and the Fourier decay problem. Positive results for any of these problems require a quantitative understanding of various geometric properties of the given measure, including curvature and arithmetic resonance. In this paper we establish a unified framework for providing negative results for all three problems (that is, we provide explicit obstructions to a measure satisfying certain Fourier restriction, $L^p$-improving, or Fourier decay estimates) by quantifying flat parts of the measure in the spirit of the well-known Knapp examples from harmonic analysis. Our main interest is in the application of these abstract results in various concrete settings where we use analytic and fractal geometric concepts to force `flatness'. Our framework applies generally and this allows us to unify and extend various parts of the literature. Some representative applications include: (i) we bound the Fourier dimension of the surface measure on a compact $C^2$ surface of arbitrary dimension above by the smallest ambient rank of a point on the surface; (ii) we prove that the Fourier dimension of a smooth curve in $\mathbb{R}^d$ is at most $4/(d+1)$ and so such curves cannot be Salem for $d \geq 4$ with analogous results for higher dimensional submanifolds; (iii) we obtain explicit upper bounds for the Fourier dimension of the Patterson-Sullivan measure for parabolic Kleinian group actions, as well as ergodic measures on self-affine sets; (iv) we establish novel connections between Fourier restriction/decay and a priori unrelated concepts in fractal geometry including the Assouad spectrum of projections and slices, and a strong form of tube-nullity. We establish several auxiliary results along the way, including a precise characterisation of L^2-flattening in terms of the Fourier spectrum.

math.CA↗

arXiv2026

Fourier restriction estimates based on $L^q$-dimensions: beyond Stein--Tomas

The well-known Stein--Tomas restriction theorem gives the sharp range of $p$ for which $L^p\to L^2$ restriction estimates hold for the surface measure on the sphere. This was generalised to arbitrary measures satisfying certain Fourier decay and Frostman conditions by Mockenhaupt, Mitsis, and Bak--Seeger, with the most general version now a fundamental result in harmonic analysis. The Frostman condition essentially asks for uniform control on the measure of small balls and is the endpoint of a continuum of more nuanced conditions which describe the local fluctuations of the measure. This analysis gives rise to the $L^q$-dimensions of a measure and these are a central concept in fractal geometry and a crucial tool in multifractal analysis and the theory of large deviations. In this paper we prove a new Fourier restriction theorem which uses the $L^q$-dimensions instead of the Frostman condition, thus providing a continuum of estimates which recover Stein--Tomas at the endpoint. Our proof gives the endpoint estimate for all values of $q\in(1,\infty]$ via Stein's complex interpolation. In particular, in the case $q=\infty$ this partially resolves a question raised by Bak and Seeger. We explore when our theorem improves on Stein--Tomas, that is, when the range is not optimised at $q=\infty$, and show that this is the case quite generally, including for certain Mandelbrot cascade measures and measures with multifractal behaviour. On the way to proving our main theorem we obtain a novel description of the $L^q$-dimensions based on certain convolution norms, which may be of interest in its own right.

math.CA↗

arXiv2026

Fourier analytic variants of the Furstenberg and Kakeya problems

We study several distinct but related Fourier analytic variants of the well-known Kakeya and Furstenberg set problems in the plane. For example, given $0<s,t<1$, we call a set $K \subseteq \mathbb{R}^2$ an $(s,t)$-Kakeya set if there exists a set of directions $E \subseteq S^1$ with Hausdorff dimension at least $t$ such that, for each $e \in E$, the set $K$ contains a subset of a unit line segment in direction $e$ whose Fourier dimension, viewed as a subset of $\mathbb{R}$, is at least $s$. For $\Delta(s,t)$ defined to be the infimum of the Fourier dimension among all $(s,t)$-Kakeya sets in $\mathbb{R}^2$, we prove that \[ \frac{2st}{s+2t} \leq \Delta(s,t) \leq \min\{s,2t\}. \] These bounds, though distinct, are asymptotically equivalent as either $s$ or $t$ tends to zero. We also obtain upper and lower bounds in the Furstenberg set version of the problem and in the case where the Hausdorff dimension of the collection of lines is replaced by the Fourier dimension.

math.CA↗

arXiv2026

Brascamp--Lieb inequalities for fractal dimensions

We use the Brascamp--Lieb inequality from functional analysis to prove novel inequalities for the upper box, packing, and Assouad dimensions of fractal sets in terms of the dimensions of certain projections. Analogous inequalities do not hold for Hausdorff or lower box dimensions. We apply these fractal Brascamp--Lieb inequalities to establish new exceptional set estimates for orthogonal projections and to provide sharp dimension estimates for certain constrained sumsets. We also establish analogous nonlinear inequalities via the nonlinear Brascamp--Lieb inequality.

math.FA↗

arXiv2026

On Fourier decay and the distance set problem

We study the Falconer distance set problem in Euclidean space and obtain improved dimensional estimates under natural Fourier analytic assumptions cast in terms of the Fourier dimension and spectrum. Interestingly, under reasonably mild assumptions, we are able to beat the $d/2$ dimension threshold in dimensions $d \geq 5$. For example, we show that (in any ambient spatial dimension $d$) a Borel set with Fourier dimension at least $2$ has a distance set of full Hausdorff dimension. We also show that (in any ambient spatial dimension $d$) a Borel set with Fourier spectrum at least $d/4+1$ at $\theta=1/2$ has a distance set of full Hausdorff dimension. In particular, this can hold for sets with Fourier dimension zero (provided $d \geq 4$). We also consider pinned variants of these problems and construct examples that demonstrate the sharpness (or near sharpness) of our results.

math.CA↗

arXiv2026

Badly approximable points on non-linear carpets

The badly approximable points in $\mathbb{R}^d$ are those for which Dirichlet's approximation theorem cannot be improved by more than a constant, that is, they are the points most difficult to approximate by rational vectors. An important problem in Diophantine approximation is to determine when the set of badly approximable points intersects a given set in full dimension. We find the first class of non-linear non-conformal attractors for which this full intersection property holds, thus answering a question of Das-Fishman-Simmons-Urba\'nski from 2019. We also provide a formula for the Hausdorff dimension of these attractors which is of independent interest.

math.NT↗

arXiv2026

An invitation to dimension interpolation

A \emph{fractal} is an object exhibiting complexity at arbitrarily small scales. In order to study and characterise fractals, one is often interested in quantifying how they fill up space on small scales. This gives rise to various notions of \emph{fractal dimension}. However, even for the simplest examples, the different definitions of dimension may completely disagree about the answer. In this expository article I will examine this phenomenon and use it to discuss and motivate \emph{dimension interpolation}. Dimension interpolation views these classical notions as boundary points of continuous families of dimensions, thus transforming isolated numerical answers into a coherent geometric picture.

math.CA↗

arXiv2026

Fourier restriction for the additive Brownian sheet

The Fourier restriction problem asks when it is meaningful to restrict the Fourier transform of a function to a given set. Many of the key examples are smooth co-dimension 1 manifolds, although there is increasing interest in fractal sets. Here we propose a natural intermediary problem where one considers the fractal surface generated by the graph of the additive Brownian sheet in $\mathbb{R}^k$. We obtain the first non-trivial estimates in this direction, giving both a sufficient condition on the range of $q\in[1,2]$ for the Fourier transform to be $L^{q}(\mathbb{R}^{k+1})\to L^2(G(W))$ bounded and a necessary condition for it to be $L^{q}(\mathbb{R}^{k+1})\to L^p(G(W))$ bounded. The sufficient condition is obtained via the Fourier spectrum, which is a family of dimensions that interpolate between the Fourier and Hausdorff dimensions. Our main technical result, which is of interest in its own right, gives a precise formula for the Fourier spectrum of the natural measure on the graph of the additive Brownian sheet, and we apply this result to the Fourier restriction problem. Our restriction estimate is stronger than the estimate obtained from the well-known Stein--Tomas restriction theorem for all $k\geq3$. We obtain the necessary condition in two different ways, one via the Fourier spectrum and one via an appropriate Knapp example.

math.PR↗

arXiv2025

$L^p$ averages of the discrete Fourier transform and applications

The discrete Fourier transform has proven to be an essential tool in many geometric and combinatorial problems in vector spaces over finite fields. In general, sets with good uniform bounds for the Fourier transform appear more `random' and are easier to analyze. However, there is a trade-off: in many cases, obtaining good uniform bounds is not possible, even in situations where many points satisfy strong pointwise bounds. To address this limitation, the first named author proposed an approach where one attempts to replace the need for uniform ($L^\infty$) bounds with suitable bounds for the $L^p$ average of the Fourier transform. In subsequent joint work, the authors applied this approach successfully to improve known results in Fourier restriction and the study of orthogonal projections. In this survey we discuss this general approach, give several examples, and exhibit some of the recent applications.

math.CO↗

arXiv2025

Sobolev and quasiconformal distortion of intermediate dimension with applications to conformal dimension

We study the distortion of intermediate dimension under supercritical Sobolev mappings and also under quasiconformal or quasisymmetric homeomorphisms. In particular, we extend to the setting of intermediate dimensions both the Gehring--V\"ais\"al\"a theorem on dilatation-dependent quasiconformal distortion of dimension and Kovalev's theorem on the nonexistence of metric spaces with conformal dimension strictly between zero and one. Applications include new contributions to the quasiconformal classification of Euclidean sets and a new sufficient condition for the vanishing of conformal box-counting dimension. We illustrate our conclusions with specific consequences for Bedford--McMullen carpets, samples of Mandelbrot percolation, and product sets containing a polynomially convergent sequence factor.

math.MG↗

arXiv2025

An improved $L^2$ restriction theorem in finite fields

Mockenhaupt and Tao (Duke 2004) proved a finite field analogue of the Stein--Tomas restriction theorem, establishing a range of $q$ for which $L^q\to L^2$ restriction estimates hold for a given measure $\mu$ on a vector space over a finite field. Their result is expressed in terms of exponents that describe uniform bounds on the measure and its Fourier transform. We generalise this result by replacing the uniform bounds on the Fourier transform with suitable $L^p$ bounds, and we show that our result improves upon the Mockenhaupt--Tao range in many cases. We also provide a number of applications of our result, including to Sidon sets and Hamming varieties.

math.CO↗

arXiv2025

Fourier analytic properties of Kakeya sets in finite fields

We prove that a Kakeya set in a vector space over a finite field of size $q$ always supports a probability measure whose Fourier transform is bounded by $q^{-1}$ for all non-zero frequencies. We show that this bound is sharp in all dimensions at least 2. In particular, this provides a new and self-contained proof that a Kakeya set in dimension 2 has size at least $q^2/2$ (which is asymptotically sharp). We also establish analogous results for sets containing $k$-planes in a given set of orientations.

math.CO↗

arXiv2025

Hausdorff dimension of restricted Kakeya sets

A Kakeya set in $\mathbb{R}^n$ is a compact set that contains a unit line segment $I_e$ in each direction $e \in S^{n-1}$. The Kakeya conjecture states that any Kakeya set in $\mathbb{R}^n$ has Hausdorff dimension $n$. We consider a restricted case where the midpoint of each line segment $I_e$ must belong to a fixed set $A$ with packing dimension at most $s \in [0, n]$. In this case, we show that the Hausdorff dimension of the Kakeya set is at least $n - s$. Furthermore, using the "bush argument", we improve the lower bound to $\max \{ n - s, n - g_n(s)\}$, where $g_n(s)$ is defined inductively. For example, when $n = 4$, we prove that the Hausdorff dimension is at least $\max\{\frac{19}{5} - \frac{3}{5}s,4-s\}$. We also establish Kakeya maximal function analogues of these results.

math.CA↗

arXiv2025

Exceptional projections in finite fields: Fourier analytic bounds and incidence geometry

We consider the problem of bounding the number of exceptional projections (projections which are smaller than typical) of a subset of a vector space over a finite field onto subspaces. We establish bounds that depend on $L^p$ estimates for the Fourier transform, improving various known bounds for sets with sufficiently good Fourier analytic properties. The special case $p=2$ recovers a recent result of Bright and Gan (following Chen), which established the finite field analogue of Peres--Schlag's bounds from the continuous setting. We prove several auxiliary results of independent interest, including a character sum identity for subspaces (solving a problem of Chen) and a full generalization of Plancherel's theorem for subspaces. These auxiliary results also have applications in affine incidence geometry, that is, the problem of estimating the number of incidences between a set of points and a set of affine $k$-planes. We present a novel and direct proof of a well-known result in this area that avoids the use of spectral graph theory, and we provide simple examples demonstrating that these estimates are sharp up to constants.

math.CO↗

arXiv2025

Applications of dimension interpolation to orthogonal projections

Dimension interpolation is a novel programme of research which attempts to unify the study of fractal dimension by considering various spectra which live in between well-studied notions of dimension such as Hausdorff, box, Assouad and Fourier dimension. These spectra often reveal novel features not witnessed by the individual notions and this information has applications in many directions. In this survey article, we discuss dimension interpolation broadly and then focus on applications to the dimension theory of orthogonal projections. We focus on three distinct applications coming from three different dimension spectra, namely, the Fourier spectrum, the intermediate dimensions, and the Assouad spectrum. The celebrated Marstrand--Mattila projection theorem gives the Hausdorff dimension of the orthogonal projection of a Borel set in Euclidean space for almost all orthogonal projections. This result has inspired much further research on the dimension theory of projections including the consideration of dimensions other than the Hausdorff dimension, and the study of the exceptional set in the Marstrand--Mattila theorem.

math.MG↗

arXiv2024

$L^2$ restriction estimates from the Fourier spectrum

The Stein--Tomas restriction theorem is an important result in Fourier restriction theory. It gives a range of $q$ for which $L^q\to L^2$ restriction estimates hold for a given measure, in terms of the Fourier and Frostman dimensions of the measure. We generalise this result by using the Fourier spectrum; a family of dimensions that interpolate between the Fourier and Sobolev dimensions for measures. This gives us a continuum of Stein--Tomas type estimates, and optimising over this continuum gives a new $L^q\to L^2$ restriction theorem which often outperforms the Stein--Tomas result. We also provide results in the other direction by giving a range of $q$ in terms of the Fourier spectrum for which $L^q\to L^2$ restriction estimates fail, generalising an observation of Hambrook and {\L}aba. We illustrate our results with several examples, including the surface measure on the cone, the moment curve, and several fractal measures.

math.CA↗

arXiv2024

On variants of the Furstenberg set problem

Given an integer $d \geq 2$, $s \in (0,1]$, and $t \in [0,2(d-1)]$, suppose a set $X$ in $\mathbb{R}^d$ has the following property: there is a collection of lines of packing dimension $t$ such that every line from the collection intersects $X$ in a set of packing dimension at least $s$. We show that such sets must have packing dimension at least $\max\{s,t/2\}$ and that this bound is sharp. In particular, the special case $d=2$ solves a variant of the Furstenberg set problem for packing dimension. We also solve the upper and lower box dimension variants of the problem. In both of these cases the sharp threshold is $\max\{s,t+1-d\}$.

math.CA↗