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Malabika Pramanik

Publications and source records attributed to Malabika Pramanik.

At least 19 recordsLinked to original sources
arXiv2026

Directional maximal operators in the plane

This monograph investigates the Lebesgue boundedness of planar directional maximal operators $D_Ω$. These are maximal averages of functions over line segments in $\mathbb R^2$ whose slopes lie in a specified set $Ω\subseteq\mathbb R$. A large body of work has identified a geometric property of $Ω$, called finite-order lacunarity, as a key factor in ensuring that $D_Ω$ is Lebesgue bounded. While several variations of this notion exist, they all centre on the distribution of gaps in $Ω$. Building on earlier work, an article of Bateman(2009) asserted a dichotomy for such operators. Namely, $D_Ω$ is bounded on $L^p$ for all $p\in (1,\infty)$ precisely when the slope set $Ω$ is finite-order lacunary, or equivalently, when $Ω$ does not admit Kakeya-type sets. Conversely, sublacunary direction sets $Ω$ admit Kakeya-like phenomena, implying that $D_Ω$ is unbounded on $L^p$ for all $p\in [1,\infty)$. Recent work of Hagelstein, Radillo-Murguia, and Stokolos(2024) identified a gap in the proof of this assertion and produced counterexamples for which the separation mechanism underlying that proof fails, demonstrating the need for a corrected framework. We establish the corrected characterization by introducing a new notion of admissible finite-order lacunarity that faithfully reflects the combinatorial structure of the direction set. This leads to a tree-theoretic characterization in terms of finite splitting number and provides the foundation for new geometric and probabilistic constructions establishing the equivalence between finite-order lacunarity, the absence of Kakeya-type sets, and the boundedness of directional maximal operators. The resulting framework not only resolves the gap in the earlier proof, but also identifies admissible finite-order lacunarity as the structural invariant governing these phenomena.

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arXiv2025

Distances in sparse sets of large Hausdorff dimension

The distance set $Δ(E)$ of a set $E$ consists of all non-negative numbers that represent distances between pairs of points in $E$. This paper studies sparse (less than full-dimensional) Borel sets in $\mathbb R^d$, $d \geq 2$ with a focus on properties of their distance sets. Our results are of four types. First, we generalize a classical result of Steinhaus (1920) to Borel sets $E \subseteq [0,1]^d$ with $s$-dimensional Hausdorff content larger than $(1 - ρ)$, for small $ρ> 0$ and $s$ close to $d$. For such sets, we show that $Δ(E) \supseteq [a, b]$, where $0<a<b$ depend only on $d$ and $ρ$. This leads to our second result, a quantitative formulation of a theorem of Mattila and Sj$\ddot{\text{o}}$lin (1999). For an arbitrary Borel set $E \subseteq [0,1]^d$ of large Hausdorff dimension, we show that $Δ(E)$ contains a union of intervals whose lengths are dictated by cubes where $E$ holds high density. This structure theorem in turn yields a tool for identifying abundance of distances; this is the third contribution of this article. It allows us to formulate a size property of a set that guarantees all sufficiently large distances, generalizing earlier work of Bourgain (1986). It can also be used to construct examples of totally disconnected sparse sets with this property. Finally, we explore special features of $Δ(E)$ if $E$ is assumed to have certain structural regularity in addition to large Hausdorff dimension. The additional regularity is harnessed via $L^2$-Fourier asymptotics of measures supported on $E$. Applications of this phenomenon give new information on $Δ(E)$ when $E$ is locally uniformly $s$-dimensional and quasi-regular, in the terminology of Strichartz (1990). A number of new examples, counterexamples and open problems are discussed.

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arXiv2024

A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in $\mathbb{R}^3$

We resolve a conjecture of Fässler and Orponen on the dimension of exceptional projections to one-dimensional subspaces indexed by a space curve in $\mathbb{R}^3$. We do this by obtaining sharp $L^p$ bounds for a variant of the Wolff circular maximal function over fractal sets for a class of $C^2$ curves related to Sogge's cinematic curvature condition. A key new tool is the use of lens cutting techniques from discrete geometry.

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arXiv2024

Measures supported on partly normal numbers

A real number $x$ is normal with respect to an integer base $b \geq 2$ if its digit expansion in this base is ``equitable'', in the sense that for $k \geq 1$, every ordered sequence of $k$ digits from $\{0, 1, \ldots, b-1\}$ occurs in the digit expansion of $x$ with the same limiting frequency. Borel's classical result \cite{b09} asserts that Lebesgue-almost every number $x$ is normal in every base $b \geq 2$. This three-part article considers sets of partial normality. Given any choice of integer bases $\mathscr{B}, \mathscr{B}' \subseteq \{2, 3, \ldots\}$, we investigate measure-theoretic properties of the set $\mathscr{N}(\mathscr{B}, \mathscr{B}')$, whose members are, by definition, normal in the bases of $\mathscr{B}$ and non-normal in the bases of $\mathscr{B}'$. A pair of sets $(\mathscr{B}, \mathscr{B}')$ is compatible if any $(b, b') \in \mathscr{B} \times \mathscr{B}'$ is multiplicatively independent. For compatible $(\mathscr{B}, \mathscr{B}')$ with $\mathscr{B}' \ne \emptyset$, we construct singular probability measures supported on $\mathscr N(\mathscr{B}, \mathscr{B}')$ that are both Frostman and Rajchman, extending prior work of Pollington \cite{p81} and Lyons \cite{l86}. The Rajchman property completely answers a question of Kahane and Salem \cite{Kahane-Salem-64}, identifying $\mathscr N(\mathscr{B}, \mathscr{B}')$ as a set of multiplicity (in the Fourier-analytic sense) if and only if $(\mathscr{B}, \mathscr{B}')$ is compatible. The methodological contribution of the article is the construction of a class of probability measures called skewed measures. These measures depend on a number of parameters that can be independently adjusted to ensure (subsets of) properties such as almost everywhere normality, non-normality, ball conditions and Fourier decay.

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arXiv2024

On odd-normal numbers

A real number $x$ is considered normal in an integer base $b \geq 2$ if its digit expansion in this base is ``equitable'', ensuring that for each $k \geq 1$, every ordered sequence of $k$ digits from $\{0, 1, \ldots, b-1\}$ occurs in the digit expansion of $x$ with the same limiting frequency. Borel's classical result \cite{b09} asserts that Lebesgue-almost every $x \in \mathbb R$ is normal in every base $b \geq 2$. This paper serves as a case study of the measure-theoretic properties of Lebesgue-null sets containing numbers that are normal only in certain bases. We consider the set $\mathscr N(\mathscr{O}, \mathscr{E})$ of reals that are normal in odd bases but not in even ones. This set has full Hausdorff dimension \cite{p81} but zero Fourier dimension. The latter condition means that $\mathscr N(\mathscr{O}, \mathscr{E})$ cannot support a probability measure whose Fourier transform has power decay at infinity. Our main result is that $\mathscr N(\mathscr{O}, \mathscr{E})$ supports a Rajchman measure $μ$, whose Fourier transform $\widehatμ(ξ)$ approaches 0 as $|ξ| \rightarrow \infty$ by definiton, albeit slower than any negative power of $|ξ|$. Moreover, the decay rate of $\widehatμ$ is essentially optimal, subject to the constraints of its support. The methods draw inspiration from the number-theoretic results of Schmidt \cite{s60} and a construction of Lyons \cite{l86}. As a consequence, $\mathscr N(\mathscr{O}, \mathscr{E})$ emerges as a set of multiplicity, in the sense of Fourier analysis. This addresses a question posed by Kahane and Salem \cite{Kahane-Salem-64} in the special case of $\mathscr N(\mathscr{O}, \mathscr{E})$.

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arXiv2023

Two-point patterns determined by curves

Let $Γ\subset \mathbb{R}^d$ be a smooth curve containing the origin. Does every Borel subset of $\mathbb R^d$ of sufficiently small codimension enjoy a Sárközy-like property with respect to $Γ$, namely, contain two elements differing by a member of $Γ\setminus \{0\}$? Kuca, Orponen, and Sahlsten have answered this question in the affirmative for a specific curve with nonvanishing curvature, the standard parabola $(t, t^2)$ in $\mathbb{R}^2$. In this article, we use the analytic notion of "functional type", a generalization of curvature ubiquitous in harmonic analysis, to study containment of patterns in sets of large Hausdorff dimension. Specifically, for $\textit{every}$ curve $Γ\subset \mathbb{R}^d$ of finite type at the origin, we prove the existence of a dimensional threshold $\varepsilon >0$ such that every Borel subset of $\mathbb{R}^d$ of Hausdorff dimension larger than $d - \varepsilon$ contains a pair of points of the form $\{x, x+γ\}$ with $γ\in Γ\setminus \{0\}$. The threshold $\varepsilon$ we obtain, though not optimal, is shown to be uniform over all curves of a given "type". We also demonstrate that the finite type hypothesis on $Γ$ is necessary, provided $Γ$ either is parametrized by polynomials or is the graph of a smooth function. Our results therefore suggest a correspondence between sets of prescribed Hausdorff dimension and the "types" of two-point patterns that must be contained therein.

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arXiv2022

Large sets avoiding affine copies of infinite sequences

A conjecture of Erdős states that for any infinite set $A \subseteq \mathbb R$, there exists $E \subseteq \mathbb R$ of positive Lebesgue measure that does not contain any nontrivial affine copy of $A$. The conjecture remains open for most fast-decaying sequences, including the geometric sequence $A = \{2^{-k} : k \geq 1\}$. In this article, we consider infinite decreasing sequences $A = \{a_k: k \geq 1\}$ in ${\mathbb R}$ that converge to zero at a prescribed rate; namely $\log (a_n/a_{n+1}) = e^{φ(n)} $, where $φ(n)/n\to 0$ as $n\to\infty$. This condition is satisfied by sequences whose logarithm has polynomial decay, and in particular by the geometric sequence. For any such sequence $A$, we construct a Borel set ${\mathcal O}\subseteq \mathbb R$ of Hausdorff dimension 1, but Lebesgue measure zero, that avoids all nontrivial affine copies of $A\cup\{0\}$.

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arXiv2021

Symmetrization of a family of Cauchy-Like kernels: Global instability

The fundamental role of the Cauchy transform in harmonic and complex analysis has led to many different proofs of its $L^2$ boundedness. In particular, a famous proof of Melnikov-Verdera [18] relies upon an iconic symmetrization identity of Melnikov [17] linking the universal Cauchy kernel $K_0$ to Menger curvature. Analogous identities hold for the real and the imaginary parts of $K_0$ as well. Such connections have been immensely productive in the study of singular integral operators and in geometric measure theory. \vskip0.1in In this article, given any function $h: \mathbb C \rightarrow \mathbb R$, we consider an inhomogeneous variant $K_h$ of $K_0$ which is inspired by complex function theory. While an operator with integration kernel $K_h$ is easily seen to be $L^2$-bounded for all $h$, the symmetrization identities for each of the real and imaginary parts of $K_h$ show a striking lack of robustness in terms of boundedness and positivity, two properties that were critical in [18] and in subsequent works by many authors. Indeed here we show that for any continuous $h$ on $\mathbb C$, the only member of $\{K_h\}_h$ whose symmetrization has the right properties is $K_0$! This global instability complements our previous investigation [12] of symmetrization identities in the restricted setting of a curve, where a sub-family of $\{K_h\}_h$ displays very different behaviour than its global counterparts considered here. Our methods of proof have some overlap with techniques in recent work of Chousionis-Prat [5] and Chunaev [6].

math.CV↗
arXiv2021

Symmetrization of a Cauchy-like kernel on curves

Given a curve $Γ\subset \mathbb C$ with specified regularity, we investigate boundedness and positivity for a certain three-point symmetrization of a Cauchy-like kernel $K_Γ$ whose definition is dictated by the geometry and complex function theory of the domains bounded by $Γ$. Our results show that $\mathtt S[\text{Re} K_Γ]$ and $\mathtt S[\text{Im} K_Γ]$ (namely, the symmetrizations of the real and imaginary parts of $K_Γ$) behave very differently from their counterparts for the Cauchy kernel previously studied in the literature. For instance, the quantities $\mathtt S[\text{Re} K_Γ](\mathbf z)$ and $\mathtt S[\text{Im} K_Γ](\mathbf z)$ can behave like $\frac32c^2(\mathbf z)$ and $-\frac12c^2(\mathbf z)$, where $\mathbf z$ is any three-tuple of points in $Γ$ and $c(\mathbf z)$ is the Menger curvature of $\mathbf z$. For the original Cauchy kernel, an iconic result of M. Melnikov gives that the symmetrized forms of the real and imaginary parts are each equal to $\frac12c^2(\mathbf z)$ for all three-tuples in $\mathbb C$.

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arXiv2020

Restriction of Laplace-Beltrami eigenfunctions to arbitrary sets on manifolds

Given a compact Riemannian manifold $(M, g)$ without boundary, we estimate the Lebesgue norm of Laplace-Beltrami eigenfunctions when restricted to a wide variety of subsets $Γ$ of $M$. The sets $Γ$ that we consider are Borel measurable, Lebesgue-null but otherwise arbitrary with positive Hausdorff dimension. Our estimates are based on Frostman-type ball growth conditions for measures supported on $Γ$. For large Lebesgue exponents $p$, these estimates provide a natural generalization of $L^p$ bounds for eigenfunctions restricted to submanifolds, previously obtained in \cite{Ho68, Ho71, Sog88, BGT07}. Under an additional measure-theoretic assumption on $Γ$, the estimates are shown to be sharp in this range. As evidence of the genericity of the sharp estimates, we provide a large family of random, Cantor-type sets that are not submanifolds, where the above-mentioned sharp bounds hold almost surely.

math.AP↗
arXiv2020

Fourier dimension and avoidance of linear patterns

The results in this paper are of two types. On one hand, we construct sets of large Fourier dimension that avoid nontrivial solutions of certain classes of linear equations. In particular, given any finite collection of translation-invariant linear equations of the form \begin{equation} \sum_{i=1}^v m_ix_i=m_0x_0, \; \text{ with } (m_0, m_1, \cdots, m_v) \in \mathbb N^{v+1}, m_0 = \sum_{i=1}^{v} m_i \text{ and } v \geq 2, \label{rational-eqn} \end{equation} we find a Salem set $E \subseteq [0,1]$ of dimension 1 that contains no nontrivial solution of any of these equations; in other words, there does not exist a vector $(x_0, x_1, \cdots, x_v) \in E^{v+1}$ with distinct entries that satisfies any of the given equations. Variants of this construction can also be used to obtain Salem sets that avoid solutions of translation-invariant linear equations of other kinds, for instance, when the collection of linear equations to be avoided is uncountable or has irrational coefficients. While such constructions seem to suggest that Salem sets can avoid many configurations, our second type of results offers a counterpoint. We show that a set in $\mathbb R$ whose Fourier dimension exceeds $2/(v+1)$ cannot avoid nontrivial solutions of all equations of the above form. In particular, a set of positive Fourier dimension must contain a nontrivial linear pattern of the above form for some $v$, and hence cannot be rationally independent. This is in stark contrast with known results \cite{M17} that ensure the existence of rationally independent sets of full Hausdorff dimension. The latter class of results may be viewed as quantitative evidence of the structural richness of Salem sets of positive dimension, even if the dimension is arbitrarily small.

math.CA↗
arXiv2020

Bergman spaces under maps of monomial type

For appropriate domains $Ω_{1}, Ω_{2}$ we consider mappings $Φ_{\mathbf A}:Ω_{1}\toΩ_{2}$ of monomial type. We obtain an orthogonal decomposition of the Bergman space $\mathcal A^{2}(Ω_{1})$ into finitely many closed subspaces indexed by characters of a finite Abelian group associated to the mapping $Φ_{\mathbf A}$. We then show that each subspace is isomorphic to a weighted Bergman space on $Ω_{2}$. This leads to a formula for the Bergman kernel on $Ω_{1}$ as a sum of weighted Bergman kernels on $Ω_{2}$

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arXiv2020

$L^2$ Bounds for a maximal directional Hilbert transform

Given any finite direction set $Ω$ of cardinality $N$ in Euclidean space, we consider the maximal directional Hilbert transform $H_Ω$ associated to this direction set. Our main result provides an essentially sharp uniform bound, depending only on $N$, for the $L^2$ operator norm of $H_Ω$ in dimensions 3 and higher. The main ingredients of the proof consist of polynomial partitioning tools from incidence geometry and an almost-orthogonality principle for $H_Ω$. The latter principle can also be used to analyze special direction sets $Ω$, and derive sharp $L^2$ estimates for the corresponding operator $H_Ω$ that are typically stronger than the uniform $L^2$ bound mentioned above. A number of such examples are discussed.

math.CA↗
arXiv2020

A proof of the Erdös similarity conjecture

We show that for any infinite set $A$ in ${\mathbb R}$, there exists a compact set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that does not contain any non-trivial affine copy of $A$. This proves the Erdös similarity conjecture.

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arXiv2019

Polynomial Roth theorems on sets of fractional dimensions

Let $E\subset \mathbb{R}$ be a closed set of Hausdorff dimension $α\in (0, 1)$. Let $P: \mathbb{R}\to \mathbb{R}$ be a polynomial without a constant term whose degree is bigger than one. We prove that if $E$ supports a probability measure satisfying certain dimension condition and Fourier decay condition, then $E$ contains three points $x, x+t, x+P(t)$ for some $t>0$. Our result extends the one of Laba and the third author to the polynomial setting, under the same assumption. It also gives an affirmative answer to a question in Henriot, Laba and the third author.

math.CA↗
arXiv2019

Large Sets Avoiding Rough Patterns

The pattern avoidance problem seeks to construct a set $X\subset \mathbb{R}^d$ with large dimension that avoids a prescribed pattern. Examples of such patterns include three-term arithmetic progressions (solutions to $x_1 - 2x_2 + x_3 = 0$), or more general patterns of the form $f(x_1, \dots, x_n) = 0$. Previous work on the subject has considered patterns described by polynomials, or by functions $f$ satisfying certain regularity conditions. We consider the case of `rough' patterns, not necessarily given by the zero-set of a function with prescribed regularity. There are several problems that fit into the framework of rough pattern avoidance. As a first application, if $Y \subset \mathbb{R}^d$ is a set with Minkowski dimension $α$, we construct a set $X$ with Hausdorff dimension $d-α$ such that $X+X$ is disjoint from $Y$. As a second application, if $C$ is a Lipschitz curve, we construct a set $X \subset C$ of dimension $1/2$ that does not contain the vertices of an isosceles triangle.

math.CA↗
arXiv2018

On the maximal directional Hilbert transform

For any dimension $n \geq 2$, we consider the maximal directional Hilbert transform $\mathscr{H}_U$ on $\mathbb R^n$ associated with a direction set $U \subseteq \mathbb S^{n-1}$: \[ \mathscr{H}_Uf(x) := \frac{1}π \sup_{v \in U} \Bigl| \text{p.v.} \int f(x - tv) \, \frac{dt}{t}\Bigr|.\] The main result in this article asserts that for any exponent $p \in (1, \infty)$, there exists a positive constant $C_{p,n}$ such that for any finite direction set $U \subseteq \mathbb S^{n-1}$, \[||\mathscr{H}_U||_{p \rightarrow p} \geq C_{p,n} \sqrt{\log \#U}, \] where $\#U$ denotes the cardinality of $U$. As a consequence, the maximal directional Hilbert transform associated with an infinite set of directions cannot be bounded on $L^p(\mathbb{R}^{n})$ for any $n\geq 2$ and any $p \in (1, \infty)$. This completes a result of Karagulyan, who proved a similar statement for $n=2$ and $p=2$.

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