Author

# Alex Iosevich

Other affiliations: Wright State University, University of Missouri, Georgetown University

Bio: Alex Iosevich is an academic researcher from University of Rochester. The author has contributed to research in topics: Vector space & Hausdorff dimension. The author has an hindex of 30, co-authored 244 publications receiving 3502 citations. Previous affiliations of Alex Iosevich include Wright State University & University of Missouri.

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TL;DR: In this article, the authors studied the Erdos/Falconer distance problem in vector spaces over finite fields and developed a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in order to provide estimates for minimum cardinality of the distance set.

Abstract: We study the Erdos/Falconer distance problem in vector spaces over finite fields. Let be a finite field with elements and take , . We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in to provide estimates for minimum cardinality of the distance set in terms of the cardinality of . Bounds for Gauss and Kloosterman sums play an important role in the proof.

182 citations

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TL;DR: Theorem 0.1. as mentioned in this paper shows that a convex compact set admits a spectrum if and only if it is possible to tile R by a family of translates of the convex set.

Abstract: Let Ω be a compact convex domain in the plane. We prove that L2(Ω) has an orthogonal basis of exponentials if and only if Ω tiles the plane by translation. 0. Introduction Let Ω be a domain in R, i.e., Ω is a Lebesgue measurable subset of R with finite non-zero Lebesgue measure. We say that a set Λ ⊂ R is a spectrum of Ω if {e}λ∈Λ is an orthogonal basis of L(Ω). Fuglede Conjecture. ([Fug74]) A domain Ω admits a spectrum if and only if it is possible to tile R by a family of translates of Ω. If a tiling set or a spectrum set is assumed to be a lattice, then the Fuglede Conjecture follows easily by the Poisson summation formula. In general, this conjecture is nowhere near resolution, even in dimension one. However, there is some recent progress under an additional assumption that Ω is convex. In [IKP99], the authors prove that the ball does not admit a spectrum in any dimension greater than one. In [Kol99], Kolountzakis proves that a nonsymmetric convex body does not admit a spectrum. In [IKT00], the authors prove that any convex body in R, d > 1, with a smooth boundary, does not admit a spectrum. In two dimensions, the same conclusion holds if the boundary is piece-wise smooth and has at least one point of non-vanishing curvature. The main result of this paper is the following: Theorem 0.1. The Fuglede conjecture holds in the special case where Ω is a convex compact set in the plane. More precisely, Ω admits a spectrum if and only if Ω is either a quadrilateral or a hexagon. Our task is simplified by the following result due to Kolountzakis. See [Kol99]. Received May 1, 2001. The research of Alex Iosevich is partially supported by the NSF Grants DMS00-87339 and DMS02-45369. The research of Nets Katz is partially supported by the NSF Grant DMS-9801410. Terry Tao is a Clay prize fellow and is supported by grants from the Packard and Sloan foundations.

136 citations

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TL;DR: In this article, it was shown that there is a point in a compact set of Hausdorff dimension greater than 5/4 where the set of distances has positive Lebesgue measure.

Abstract: If $$E \subset \mathbb {R}^2$$ is a compact set of Hausdorff dimension greater than 5 / 4, we prove that there is a point $$x \in E$$ so that the set of distances $$\{ |x-y| \}_{y \in E}$$ has positive Lebesgue measure.

128 citations

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Rutgers University

^{1}, University of Rochester^{2}, Michigan State University^{3}, University of Bristol^{4}TL;DR: In this article, a pointwise and average bound for the number of incidences between points and hyperplanes in vector spaces over finite fields has been established, and the Erdos-Falconer distance conjecture does not hold in this setting.

Abstract: We prove a pointwise and average bound for the number of incidences between points and hyperplanes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering F q , the finite field with q elements, by A · A + ··· + A · A, where A is a subset F q of sufficiently large size. We also use the incidence machinery and develop arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting. On the positive side, we obtain good exponents for the Erdos-Falconer distance problem for subsets of the unit sphere in F d q and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher-dimensional vector spaces over general finite fields.

109 citations

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TL;DR: In this article, the authors improved the exponent of the Falconer distance problem in the finite field setting to Ω(d+1/Ωd+2/2d-1) using Fourier analytic methods and showed that this exponent is sharp in odd dimensions.

Abstract: 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 \({\frac{d+k}{2}}\).

109 citations

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28,685 citations

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TL;DR: 5. M. Green, J. Schwarz, and E. Witten, Superstring theory, and An interpretation of classical Yang-Mills theory, Cambridge Univ.

Abstract: 5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

1,252 citations