Falconer Distance Research Articles

Micromeasure distributions and applications for conformally generated fractals

AbstractWe study the scaling scenery of Gibbs measures for subshifts of finite type on self-conformal fractals and applications to Falconer's distance set problem and dimensions of projections. Our analysis includes hyperbolic Julia sets, limit sets of Schottky groups and graph-directed self-similar sets.

Results on the Erdős–Falconer distance problem in [formula omitted] for odd q

The Erdős–Falconer distance problem in Zqd asks one to show that if E⊂Zqd is of sufficiently large cardinality, then the set of distances determined by E satisfies Δ(E)=Zq. Previous results were known only in the case q=pℓ, where p is an odd prime, and as such only showed that all units were obtained in the distance set. We give the first such result over rings Zq where q is no longer confined to be a prime power, and despite this, we show that the distance set of E contains all of Zq whenever E is of sufficiently large cardinality.

Scaling scenery of [formula omitted] invariant measures

We study the scaling scenery and limit geometry of invariant measures for the non-conformal toral endomorphism (x,y)↦(mxmod1,nymod1) that are Bernoulli measures for the natural Markov partition. We show that the statistics of the scaling can be described by an ergodic CP-chain in the sense of Furstenberg. Invoking the machinery of CP-chains yields a projection theorem for Bernoulli measures, which generalises in part earlier results by Hochman–Shmerkin and Ferguson–Jordan–Shmerkin. We also give an ergodic theoretic criterion for the dimension part of Falconer's distance set conjecture for general sets with positive length using CP-chains and hence verify it for various classes of fractals such as self-affine carpets of Bedford–McMullen, Lalley–Gatzouras and Barański class and all planar self-similar sets.

Theory of Dimension for Large Discrete Sets and Applications

We define two notions of discrete dimension based on the Minkowski and Hausdorff dimensions in the continuous setting. After proving some basic results illustrating these definitions, we apply this machinery to the study of connections between the Erdős and Falconer distance problems in geometric combinatorics and geometric measure theory, respectively.

On the generalized Erdős–Falconer distance problems over finite fields

In this paper we study the generalized Erdős–Falconer distance problems in the finite field setting via spectra of directed graph. The generalized distances are defined in terms of polynomials, and various formulas for sizes of distance sets are obtained. In particular, we give graph-theoretic proofs for various results due to Koh and Shen (2011). As a consequence, we also obtain a better bound under a weaker condition of the restricted sets.

On the Erdős–Falconer distance problem for two sets of different size in vector spaces over finite fields

We consider a finite fields version of the Erdős–Falconer distance problem for two different sets. In a certain range for the sizes of the two sets we obtain results of the conjectured order of magnitude.

The generalized Erdős–Falconer distance problems in vector spaces over finite fields

In this paper we study the generalized Erdos-Falconer distance problems in the finite field setting. The generalized distances are defined in terms of polynomials, and various formulas for sizes of distance sets are obtained. In particular, we develop a simple formula for estimating the cardinality of distance sets determined by diagonal polynomials. As a result, we generalize the spherical distance problems due to Iosevich and Rudnev and the cubic distance problems due to Iosevich and Koh. Moreover, our results are of higher dimensional version for Vu's work on two dimension. In addition, we set up and study the generalized pinned distance problems in finite fields. We give a nice generalization of some recent work in which the pinned distance problems related to spherical distances were investigated. Discrete Fourier analysis and exponential sum estimates play an important role in our proof.

Sharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields

In this paper we study extension theorems associated with general varieties in two dimensional vector spaces over finite fields. Applying Bezout’s theorem, we obtain the sufficient and necessary conditions on general curves where sharp L^p - L^r extension estimates hold. Our main result can be considered as a nice generalization of works by Mockenhaupt and Tao in [17] and Iosevich and Koh in [10]. As an application of our sharp extension estimates, we also study the Falconer distance problems in two dimensions.

Fourier integral operators, fractal sets, and the regular value theorem

We prove that if E ⊂ R 2 d , for d ⩾ 2 , is an Ahlfors–David regular product set of sufficiently large Hausdorff dimension, denoted by dim H ( E ) , and ϕ is a sufficiently regular function, then the upper Minkowski dimension of the set { w ∈ E : ϕ l ( w ) = t l ; 1 ⩽ l ⩽ m } does not exceed dim H ( E ) − m , in line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier integral operators and are intimately connected with the Falconer distance conjecture in geometric measure theory. We shall see that our results are, in general, sharp in the sense that if the Hausdorff dimension is smaller than a certain threshold, then the dimensional inequality fails in a quantifiable way. The constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry.

Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold α > 0 such that \({|\Delta(E)| \gtrsim q}\) whenever \({|E| \gtrsim q^{\alpha}}\), where \({E \subset {\mathbb {F}}_q^d}\), the d-dimensional vector space over a finite field with q elements (not necessarily prime). Here \({\Delta(E)=\{{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2: x,y \in E\}}\). Iosevich and Rudnev (Trans Am Math Soc 359(12):6127–6142, 2007) established the threshold \({\frac{d+1}{2}}\), and in Hart et al. (Trans Am Math Soc 363:3255–3275, 2011) proved that this exponent is sharp in odd dimensions. In two dimensions we improve the exponent to \({\tfrac{4}{3}}\), consistent with the corresponding exponent in Euclidean space obtained by Wolff (Int Math Res Not 10:547–567, 1999). The pinned distance set \({\Delta_y(E)=\{{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2: x\in E\}}\) for a pin \({y\in E}\) has been studied in the Euclidean setting. Peres and Schlag (Duke Math J 102:193–251, 2000) showed that if the Hausdorff dimension of a set E is greater than \({\tfrac{d+1}{2}}\), then the Lebesgue measure of Δy(E) is positive for almost every pin y. In this paper, we obtain the analogous result in the finite field setting. In addition, the same result is shown to be true for the pinned dot product set \({\Pi_y(E)=\{x\cdot y: x\in E\}}\). Under the additional assumption that the set E has Cartesian product structure we improve the pinned threshold for both distances and dot products to \({\frac{d^2}{2d-1}}\). The pinned dot product result for Cartesian products implies the following sum-product result. Let \({A\subset \mathbb F_q}\) and \({z\in \mathbb F^*_q}\). If \({|A|\geq q^{\frac{d}{2d-1}}}\) then there exists a subset \({E'\subset A\times \dots \times A=A^{d-1}}\) with \({|E'|\gtrsim |A|^{d-1}}\) such that for any \({(a_1,\dots, a_{d-1}) \in E'}\), $$ |a_1A+a_2A+\dots +a_{d-1}A+zA| > \frac{q}{2}$$ where \({a_j A=\{a_ja:a \in A\},j=1,\dots,d-1}\). A generalization of the Falconer distance problem is to determine the minimal α > 0 such that E contains a congruent copy of a positive proportion of k-simplices whenever \({|E| \gtrsim q^{\alpha}}\). Here the authors improve on known results (for k > 3) using Fourier analytic methods, showing that α may be taken to be \({\frac{d+k}{2}}\).

The Erdös–Falconer Distance Problem on the Unit Sphere in Vector Spaces Over Finite Fields

Hart, Iosevich, Koh, and Rudnev [Averages over Hyperplanes, Sum-Product Theory in Vector Spaces over Finite Fields and the Erdos–Falconer Distance Conjecture, arXiv:0707.3473v2, 2007] show, using Fourier analysis method, that the finite Erdos–Falconer distance conjecture holds for subsets of the unit sphere in 𝔽qd. In this article, we give a graph theoretic proof of this result.

Generalized incidence theorems, homogeneous forms and sum–product estimates in finite fields

In recent years, sum–product estimates in Euclidean space and finite fields have received great attention. They can often be interpreted in terms of Erdős type incidence problems involving the distribution of distances, dot products, areas, and so on, which have been studied quite extensively by way of combinatorial and Fourier analytic techniques. We use both kinds of techniques to obtain sharp or near-sharp results on the distribution of volumes (as examples of d -linear homogeneous forms) determined by sufficiently large subsets of vector spaces over finite fields and the associated arithmetic expressions. Arithmetic–combinatorial techniques turn out to be optimal for dimension d ≥ 4 to this end, while for d = 3 they have failed to provide us with a result that follows from the analysis of exponential sums. To obtain the latter result we prove a relatively straightforward function version of an incidence results for points and planes previously established in [D. Hart, A. Iosevich, Sums and products in finite fields: An integral geometric viewpoint, in: Radon Transforms, Geometry, and Wavelets, Contemp. Math. 464 (2008); D. Hart, A. Iosevich, D. Koh, M. Rudnev, Averages over hyperplanes, sum–product theory in vector spaces over finite fields and the Erdős–Falconer distance conjecture, arXiv:math/0711.4427, preprint 2007]. More specifically, we prove that if E = A × ⋯ × A is a product set in F q d , d ≥ 4 , the d -dimensional vector space over a finite field F q , such that the size | E | of E exceeds q d 2 (i.e. the size of the generating set A exceeds q ) then the set of volumes of d -dimensional parallelepipeds determined by E covers F q . This result is sharp as can be seen by taking A = F p , a prime sub-field of its quadratic extension F q , with q = p 2 . For in three dimensions, however, we are able to establish the same result only if | E | ≳ q 15 8 (i.e., | A | ≥ C q 5 8 , for some C ; in fact, the q 15 8 bound can be justified for a slightly wider class of “Cartesian product-like” sets), and this uses Fourier methods. Yet we do prove a weaker near-optimal result in three dimensions: that the set of volumes generated by a product set E = A × A × A covers a positive proportion of F q if | E | > q 3 2 (so | A | > q ). Besides, without any assumptions on the structure of E , we show that in three dimensions the set of volumes covers a positive proportion of F q if | E | ≥ C q 2 , which is again sharp up to the constant C , as taking E to be a 2-plane through the origin shows.

FREIMAN THEOREM, FOURIER TRANSFORM AND ADDITIVE STRUCTURE OF MEASURES

AbstractWe use the Freiman theorem in arithmetic combinatorics to show that if the Fourier transform of certain measures satisfies sufficiently bad estimates, then the support of the measure possesses an additive structure. The result is then discussed in light of the Falconer distance problem.

The Erdös–Falconer Distance Problem, Exponential Sums, and Fourier Analytic Approach to Incidence Theorems in Vector Spaces over Finite Fields

We study the Erdös–Falconer distance problem in vector spaces over finite fields with respect to the cubic metric. Estimates for discrete Airy sums and Adolphson–Sperber estimates for exponential sums in terms of Newton polyhedra play a crucial role. Similar techniques are used to study the incidence problem between points and cubic and quadratic curves. As a result we obtain a nontrivial range of exponents that appear to be difficult to attain using combinatorial methods.

On the Mattila Integral Associated with Sign Indefinite Measures

In order to quantitatively illustrate the rôle of positivity in the Falconer distance problem, we construct a family of sign indefinite, compactly supported measures in \({\Bbb R}^d\), such that their Fourier transform and Fourier energy of dimension \(s \in (0, d)\) are uniformly bounded. However, the Mattila integral, associated with the Falconer distance problem for these measures is unbounded in the range \(0 < s < \frac{d^2}{2d-1}\).

On Falconer’s Distance Set Conjecture

In this paper, using a recent parabolic restriction estimate of Tao, we obtain improved partial results in the direction of Falconer's distance set conjecture in dimensions d\geq 3 .

Real Analysis, Harmonic Analysis and Applications to PDE

Major areas and results represented at the workshop are: I. Methods in harmonic analysis (a) Multilinear analysis: this is an outgrowth of the method of “tile decomposition” which has been so successful in solving the problems of the bilinear Hilbert transform. Recent progress involves the control of maximal trilinear operators and an extension of the Carleson–Hunt theorem. (b) Geometry of sets in \R^d : This includes recent progress on the interaction of Fourier analysis and geometric combinatorics related to the Falconer distance problem. (c) Singular integrals: A break trough has been obtained on singular integrals on solvable Iwasawa AN - groups, which require a new type of Calderón–Zygmund decomposition, since the underlying spaces have exponential volume growth. Further significant progress involves the theory of operator-valued Calderón–Zygmund operators and their connection to maximal regularity of evolution equations. (d) Oscillatory integrals, Fourierintegral operators and Maximal operators: This includes estimates of maximal operators related to polynomial polyhedra and their relations with higher dimensional complex analysis, sharp estimates for maximal operators associated to hypersurfaces in \R^3, estimates for degenerate Radon transforms and linear and bilinear estimates for oscillatory integral operators, as well as optimal Sobolev regularity for Fourier integral operators. II. Applications to P.D.E. (a) Dispersive linear and non-linear equations: Far reaching new approaches to dispersive estimates for Schrödinger equations via coherent state decompositions and a related new phase space transform adapted to the wave operator were introduced. Further significant progress includes L^p - estimates respectively blow up rates for eigenfunctions and quasimodes of elliptic operators on compact manifolds with and without boundary, and related problems for globally elliptic pseudodifferential operators, and well-posedness of the periodic KP-I equations. All these results are based in part on important ideas in harmonic analysis (such as I(d) above). (b) Schrödinger operators with rough potentials: This includes quantitative unique continuation theorems and their relations with spectral properties and the study of embedded eigenvalues of Schrödinger operators. The meeting was attended by 50 participants. The official program consisted of 23 lectures, and left sufficiant room for further activities, such as self-organised sessions and discussions among groups of participants. The organisers made an effort to include young mathematicians, and greatly appreciate the new joint program of the Oberwolfach Institute and the american NSF, which allowed to invite several outstanding young scientists from the United States.

Circular averages and Falconer/Erdös distance conjecture in the plane for random metrics

We study a variant of the Falconer distance problem for perturbations of the Euclidean and related metrics. We prove that Mattila’s criterion, expressed in terms of circular averages, which would imply the Falconer conjecture, holds on average. We also use a technique involving diophantine approximation to prove that the well-distributed case of the Erdös Distance Conjecture holds for almost every appropriate perturbation of the Euclidean metric.