Author

# Eyvindur A. Palsson

Other affiliations: University of Rochester, Williams College

Bio: Eyvindur A. Palsson is an academic researcher from Virginia Tech. The author has contributed to research in topics: Mathematics & Hausdorff dimension. The author has an hindex of 9, co-authored 53 publications receiving 287 citations. Previous affiliations of Eyvindur A. Palsson include University of Rochester & Williams College.

##### Papers

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39 citations

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TL;DR: In this paper, the authors obtained nontrivial exponents for Erd\H os-Falconer type problems using a group-theoretic method that sheds new light on the classical approach to these problems.

Abstract: We obtain nontrivial exponents for Erd\H os-Falconer type problems. Let $T_k(E)$ denote the set of distinct congruent $k$-dimensional simplexes determined by $(k+1)$-tuples of points from $E$. We prove that there exists $s_0(d) s_0(d)$, then the ${k+1 \choose 2}$-dimensional Lebesgue measure of $T_k(E)$ is positive. Results were previously obtained for triangles in the plane \cite{GI12} and in higher dimensions \cite{GGIP12}. In this paper, we improve upon those exponents, using a group-theoretic method that sheds new light on the classical approach to these problems. The key to our approach is a group action perspective which leads to natural and effective formulae related to the classical Mattila integral.

38 citations

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TL;DR: In this article, the authors study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case and provide a general mechanism to study Falconer-type problems involving $(k+1)$-point configurations in geometric measure theory, including the distribution of simplices, volumes and angles determined by the points of fractal subsets.

Abstract: We study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case. These provide a general mechanism to study Falconer-type problems involving $(k+1)$-point configurations in geometric measure theory, with $k \ge 2$, including the distribution of simplices, volumes and angles determined by the points of fractal subsets $E \subset {\Bbb R}^d$, $d \ge 2$. If $T_k(E)$ denotes the set of noncongruent $(k+1)$-point configurations determined by $E$, we show that if the Hausdorff dimension of $E$ is greater than $d-\frac{d-1}{2k}$, then the ${k+1 \choose 2}$-dimensional Lebesgue measure of $T_k(E)$ is positive. This compliments previous work on the Falconer conjecture (\cite{Erd05} and the references there), as well as work on finite point configurations \cite{EHI11,GI10}. We also give applications to Erd\"os-type problems in discrete geometry and a fractal regular value theorem, providing a multilinear framework for the results in \cite{EIT11}.

35 citations

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TL;DR: For functions F,G on R n, any k-dimensional affine subspace HR n, and p,q,r � 2 with 1 + 1+1 + 1 = 1, one has the estimate as mentioned in this paper.

Abstract: For 1 � k < n, we prove that for functions F,G on R n , any k-dimensional affine subspace HR n , and p,q,r � 2 with 1 + 1 + 1 = 1, one has the estimate ||(FG)|H||Lr(H) � ||F||�H,p (Rn) · ||G||�H,q (Rn),

29 citations

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TL;DR: In this paper, it was shown that if the Hausdorff dimension of a compact subset of a set of points has a positive Lebesgue measure, then the set of angles determined by triples of points from this set has positive lebesgue measures.

Abstract: We prove that if the Hausdorff dimension of a compact subset of ${\mathbb R}^d$ is greater than $\frac{d+1}{2}$, then the set of angles determined by triples of points from this set has positive Lebesgue measure. Sobolev bounds for bi-linear analogs of generalized Radon transforms and the method of stationary phase play a key role. These results complement those of V. Harangi, T. Keleti, G. Kiss, P. Maga, P. Mattila and B. Stenner in (\cite{HKKMMS10}). We also obtain new upper bounds for the number of times an angle can occur among $N$ points in ${\mathbb R}^d$, $d \ge 4$, motivated by the results of Apfelbaum and Sharir (\cite{AS05}) and Pach and Sharir (\cite{PS92}). We then use this result to establish sharpness results in the continuous setting. Another sharpness result relies on the distribution of lattice points on large spheres in higher dimensions.

19 citations

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01 Jan 1994

TL;DR: In this paper, the Fourier integral operators (FIFO) were examined for hyperbolic types of elliptic differential equations, and a wider class of operators, the so-called FIFO-integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Abstract: The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

582 citations

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TL;DR: In this article, it was shown that if the system of matrices B ≥ (m − n) is non-degenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of m depending on n and k, the set E contains a translate of a non-trivial k-point configuration.

Abstract: Let E ⊆ ℝ
n
be a closed set of Hausdorff dimension α. For m > n, let{B
1, …, B
k
} be n × (m − n) matrices. We prove that if the system of matrices B
j
is non-degenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of m depending on n and k, the set E contains a translate of a non-trivial k-point configuration {B
1
y, …, B
k
y}. As a consequence, we are able to establish existence of certain geometric configurations in Salem sets (such as parallelograms in ℝ
n
and isosceles right triangles in ℝ2). This can be viewed as a multidimensional analogue of the result of [25] on 3-term arithmetic progressions in subsets of ℝ.

57 citations

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TL;DR: In this article, the authors improved Peres-Schlag's result on the Pinned Distance Problem by reducing the problem to an integral where spherical averages apply, and showed that spherical average estimates imply the same dimensional threshold on both the Pinched Distance Problem and the Falconer distance problem.

Abstract: Let $${\mu}$$
be a Frostman measure on $${E\subset\mathbb{R}^d}$$
. The spherical average estimate
$$\int_{S^{d-1}}|\widehat{\mu}(r\omega)|^2\,d\omega\lesssim r^{-\beta}$$
was originally used to attack Falconer distance conjecture, via Mattila’s integral. In this paper we consider the pinned distance problem, a stronger version of Falconer distance problem, and show that spherical average estimates imply the same dimensional threshold on both of them. In particular, with the best known spherical average estimates, we improve Peres–Schlag’s result on pinned distance problem significantly. The idea in our approach is to reduce the pinned distance problem to an integral where spherical averages apply. The key new ingredient is the following identity. Using a group action argument, we show that for any Schwartz function f on $${\mathbb{R}^d}$$
and any $${x\in\mathbb{R}^d}$$
,
$$\int_0^\infty|\omega_t*f(x)|^2\,t^{d-1}dt\,=\int_0^\infty|\widehat{\omega_r}*f(x)|^2\,r^{d-1}dr,$$
where $${\omega_r}$$
is the normalized surface measure on $${r S^{d-1}}$$
. An interesting remark is that the right hand side can be easily seen equal to
$$c_d\int\left|D_x^{-\frac{d-1}{2}}e^{-2\pi it\sqrt{-\Delta}}f(x)\right|^2\,dt=c_d'\int\left|D_x^{-\frac{d-2}{2}}e^{2\pi{it}\Delta}f(x)\right|^2\,dt.$$
An alternative derivation of Mattila’s integral via group actions is also given in the Appendix.

46 citations