Problems I am currently working on

I am currently interested in the following types of problems and connections among them. There is a need for the obvious disclaimer that breaking these problems up into different categories in inherently artificial and is being done solely for the purposes of exposition. I should also mention that while classical analysis problems like bounds for maximal operators, Lp-improving measures, estimates for oscillatory integrals are not explicitly listed below, they appear, directly or indirectly, in most if not all of the papers listed below. Even in the discrete context, Fourier analysis is always lurking, however subtly.

(1) Multi-linear generalized Radon transform and applications to geometric measure theory and geometric combinatorics.

The following are my recent papers in this research direction. The first ([GGIP]), second ([IMP]) and the fourth ([GI]) listed papers involve the application of multi-linear estimates to geometric problems. An interested reader who wishes to understand the method may want to start with the fourth listed paper, co-authored with Allan Greenleaf. The most general of these publications is the first listed paper where an attempt for comprehensive treatment of the subject.

The third listed paper ([EHI]) uses projection estimate to obtain result for finite point configurations in fractal subset of the Euclidean space. Some, but not all of the exponents in this paper are improved in the first listed publication ([GGIP]). The reader may wish to compare and contrast the methods used.

The fifth listed paper studies the distribution of volumes determined by d points and the origin in fractal subsets of the d-dimensional Euclidean space. The complication here is that the problem is not translation invariant. In the translation invariant setting, where volumes are determined by d+1 points in d-dmensional space, good exponents are obtained in the first listed publication ([GGIP]).

The sixth and seventh listed publications are strongly related on the level of techniques. In ([IMS]) we prove that fractal subsets of sufficiently lage Hausdorff dimension determine a positive proportion of all possible directions. In ([IMT]) we give a simple and a natural extension of the result due to Mattila and Sjolin which shows that if the sum of Hausdorff dimensions of two subsets of d-dimensional Euclidean space exceeds d+1, then the set of distances from one set to the other contains an interval.

Restricted convolution inequalities, multilinear operators and applications, ([GGIPS]), with D. Geba, A. Greenleaf, E. Palsson and E. Sawyer

Multi-linear generalized Radon transforms and point configurations, ([GGIP]) with L. Grafakos, A. Greenleaf and E. Palsson

On angles determined by fractal subsets of the Euclidean space via Sobolev bounds for bi-linear operators, ([IMP]) with  M. Mourgoglou and E. Palsson

Multi-parameter projection theorems with applications to sums-products and finite point configurations in the Euclidean setting, ([EHI]) with B. Erdoğan and D. Hart

On three point configurations determined by subsets of the Euclidean plane, the associated bilinear operator and applications to discrete geometry, ([GI]) with A. Greenleaf

On volumes determined by subsets of Euclidean space, ([GIM]) with A. Greenleaf and M. Mourgoglou

On sets of directions determined by subsets of ${\Bbb R}^d$, ([IMS]) with M. Mourgoglou and S.Senger

On the Mattila-Sjolin theorem for distance sets, ([IMT]) M. Mourgoglou and K. Taylor

Fourier integral operators, fractal sets and the regular value theorem, ([EIT]), with S. Eswarathasan and K. Taylor

(2) Geometric combinatorics and additive number theory in Euclidean space and vector spaces over finite fields

This subject matter is closely related to the one above, with significant overlap in techniques. The first, second, third and fourth papers above can easily be listed in this category. An interested reader is encouraged to think about precisely which Euclidean techniques fail in the discrete and finite field settings, and vice versa. The following are my recent papers in this direction. All of these papers, in one form or another, deal with Erdos type problems in vector spaces over real number or over finite fields. The first ([IRZ]) and third ([IRR]) papers deal with areas of triangles determined by subsets of planes over real numbers and finite fields, respectively. The former proves a finite field variant of the celebrated Beck lemma, whereas the second uses the Elekes-Sharir paradigma and Guth-Katz incidence theory to prove a sharp result about areas of triangles in the Euclidean plane.

The second ([CIP]), sixth ([CEHIK]), eighth ([CHIKR]), ninth ([HIKR]), tenth ([IRU]), eleventh ([IH1]), twelfth ([IH2]), fourteenth ([IK]) and fifteenth ([IR]) papers deal with finite point configurations in vector spaces over finite fields, up to congruence. An interested reader is encouraged to read the last listed paper ([IR]) first as it is the one that started it all and gives a clear idea of the initial ideas that were further developed in subsequent papers.

The fourth listed paper gives a simple description of directions determined by subsets of vector spaces over finite fields using both a Fourier analytic and algebraic arguments. This paper can be viewed as a finite field analog of ([IMS]) above.

The fifth listed paper provides a series of connections between discrete, continuous and lattice points problems. In particular, the sharpness of the classical Falconer estimate from geometric measure theory is proved in higher dimensions using an arithmetic examples based on properties of paraboloids. It is based, in part, on a beautiful idea due to Pavel Valtr in the context of incidence theory.

The tenth listed paper is a systematic study of the conversion mechanism, developed about ten years ago by Steve Hofmann, Izabella Laba, Misha Rudnev and I (see ([HI], [IL], [IR2] below) in a variety of contexts. The idea is to start out with an analytic estimate and deduce an incidence type result for discrete point sets. Some interesting complications arise, many of which are still wide open.

The thirteenth listed paper provides the first known quantitative bound for sums and products in finite fields. Kloosterman sum technology is used to obtain these quantitative bounds. The bounds have since been improved and extended in variety of contexts by many authors.

The seventh listed paper ([IJL]) shows that one can obtain a non-trivial incidence theorem for any homogeneous point set and a family of surfaces, provided that the associated operator satisfying non-trivial Sobolev bounds. This is still a largely unexplored frontier and I expect much work in this direction in the coming years.

Areas of triangles and Beck's theorem in planes over finite fields, ([IRZ]) with M. Rudnev and Y.Zhai

Geometric configurations in the ring of integers modulo $p^{\ell}$, ([CIP]) with D. Covert and J. Pakianathan

On an application of Guth-Katz theorem, ([IRR]) with O. Roche-Newton and M. Rudnev

On directions determined by subsets of vector spaces over finite fields, ([IMP]) with H. Morgan and J. Pakianathan

Sharpness of Falconer's estimate in continuous and arithmetic settings, geometric incidence theorems and distribution of lattice points in convex domains, ([IS]) with S. Senger

Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates, ([CEHIK]) with J. Chapman, B. Erdogan, D. Hart, and D. Koh

Geometric incidence theorems via Fourier analysis, ([IJL]) with H. Jorati and I. Laba

Generalized incidence theorems, homogeneous forms, and sum-product estimates in finite fields, ([CHIKR]) with D. Covert, D. Hart, D. Koh, and M. Rudnev

Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture, ([HIKR]) with D. Hart, D. Koh and M. Rudnev

Theory of dimension for large discrete sets and applications, ([IRU]) with M. Rudnev and Ignacio Uriarte-Tuero

Sums and products in finite fields: an integral geometric viewpoint, ([IH1]) with D. Hart

Ubiquity of simplices in subsets of vector spaces over finite fields, ([IH2]) with D. Hart

Sum-product estimates in finite fields, ([IHS]) with D. Hart and J. Solymosi

Erdos-Falconer distance problem, exponential sums, and Fourier analytic approach to incidence theorems in vector spaces over finite fields, ([IK]) with D. Koh

Erdos distance problem in vector spaces over finite fields, ([IR]) with M. Rudnev

Falconer conjecture in the plane for random metrics, ([HI]), with S. Hofmann

K-distance sets, Falconer conjecture, and discrete analogs, ([IL]), with I. Laba

On distance measures for well-distributed sets, ([IR2]), with M. Rudnev

(3) Distribution of lattice points in convex domains and irregularity of distribution.

One of the most beautiful problems in mathematics, in my view, is the question of counting lattice points inside a large dilate of a convex domain. The Gauss Circle Problem asks for the best possible estimate on the difference between the number of lattice points inside the disk of radius R in the plane and its area. Related questions also keep popping up in harmonic analysis, mathematical physics, geometric measure theory, geometric combinatorics and many other areas of mathematics. Irregularity of distribution, which deals with similar questions for general point sets has a time honored history going back to the pioneering work of Klaus Roth in the 50s and even further.

The third ([ISS2]), fourth ([ISS1]), fifth ([BHI]) and sixth ([BIT]) listed papers are on fairly classical results where the error term in the lattice point expansion is averaged in some fashion, either by rotating the domain or by dilating the stretching parameter. The second listed publication ([IKo1]) is a discrepancy problem for the needle on the checkerboard. In the first listed paper ([IT]), Fourier integral operator technology is applied to obtain results on the number of lattice points in thin annuli near families of manifolds in Euclidean space. Non-isotropic dilations are also addressed, with connections to parabolic examples in geometric combinatorics, as in ([IS]) above.

Lattice points close to families of surfaces, non-isotropic dilations and regularity of generalized Radon transforms, ([IT]) with K. Taylor

The discrepancy of a needle on a checkerboard, II, ([IKo1]), with M. Kolountzakis

Mean lattice point discrepancy bounds, II: Convex domains in the plane, ([ISS2]), with E. Sawyer and A. Seeger

Mean square discrepancy bounds for the number of lattice points in large convex bodies, ([ISS1]), with E. Sawyer and A. Seeger

Sharp rate of average decay of the Fourier transform of a bounded set, ([BHI]), with L. Brandolini and S. Hofmann

Spherical L1- averages of the Fourier transform of the characteristic function of a convex set and the geometry of the Gauss map, ([BIT]) with L. Brandolini and G. Travaglini

(4) Exponential bases and frames and the associated combinatorics

The subject area that I keep on coming back to over the past fifteen years or so is the field of exponential bases and frames. Connections between the existence of frames and tiling, while disproved in general by Tao, Kolountzakis and Matolci, continues to be an intriguing concept, and connections with combinatorial Erdos problems keep on arising in new and unexpected ways.

The seventh ([IKP]) and eighth ([IKT1]) listed papers show that if a domain has the point of curvature on the boundary, then the L2 space of this domain does not possess an orthogonal basis of exponentials. In fact, ([IKP]) provides a connection between the problem of the existence of the exponential bases with the Erdos distance problem in geometric combinatorics. The ninth listed paper ([IKT2]) proves the Fuglede conjecture for convex planar domains, namely that the L2 of the domain has an orthogonal basis of exponentials if and only if the domain tiles the plane by translation. While the Fuglede conjecture has in general been disproved, as we mention above, it does hold for convex domains in the plane and, most likely, in my view, in higher dimensions as well.

The first two listed papers deal with the question of how many orthogonal exponentials exist in the L2 space of a domain with a smooth boundary and non-vanishing curvature. In the first listed paper ([IR3]) we prove that if the dimension is not congruent to 1 mod 4, the number is finite. If the dimension is congruent to 1 mod 4, the number must be infinite, but the vectors generating the exponentials must lie on a line. In the second listed paper ([IKo2]), we prove, improving upon a previous result with Philippe Jaming, that in the disk of radius R, the number of such exponentials is at most CR^{2/3}.

The fourth ([IKo3]) and the fifth ([IP1]) listed papers are dedicated to the understanding of the distribution of frames on domains in Euclidean space. Weyl type formulas are proved, with fractal geometry playing an interesting role.

The sixth ([IP2]) paper, one of my favorite projects of all time, we prove that a set of vectors is a tiling set of the unit cube by translation if and only if this set generates the exponential basis for L2 space of the cube. This result was proved simultaneously, by different methods, by Lagarias, Reed and Wang.

A combinatorial approach to orthogonal exponentials, ([IR3]), with M. Rudnev.

Size of orthogonal sets of exponentials for the disk, ([IKo2]), with M. Kolountzakis

Periodicity of the spectrum in dimension one, ([Iko3]), with M. Kolountzakis

A Weyl type formula for Fourier spectra and frames, ([Iko4]), with M. Kolountzakis

How large are the spectral gaps? ([IP1]), with S. Pedersen

Spectral and Tiling properties of the Unit Cube, ([IP2]), with S. Pedersen

Fourier bases and a distance problem of Erd\H os, ([IKP]), with N. Katz and S. Pedersen

Convex bodies with a point of curvature do not have Fourier bases, ([IKT1]), with N. Katz and T. Tao

Fuglede conjecture holds for convex planar domains, ([IKT2]), with N. Katz and T. Tao