A. Iosevich, H. Jorati, I. Laba
We prove that non-trivial bounds for generalized Radon transforms imply correspondingly non-trivial discrete incidence theorems for manifolds and suitably regular point sets.
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We prove several incidence theorems in vector spaces over finite fields using bounds for various classes of exponential sums and apply these to Erdos-Falconer type distance problems.
Exceptional projections in finite fields: Fourier analytic bounds and incidence geometry
本文研究有限域上子集例外投影数量的上界，通过傅里叶$L^p$估计改进已知结果，并给出子空间特征和恒等式及Plancherel定理推广，应用于仿射关联几何，提供尖锐性证明。
We improve the range for the discrete Fourier restriction to the four and five dimensional spheres. We rely on two new ingredients, incidence theory and Siegel's mass formula.
In this paper we prove, for all $d \ge 2$, that for no $s<\frac{d+1}{2}$ does $I_s(μ)<\infty$ imply the canonical Falconer distance problem incidence bound, or the analogous estimate where the Euclidean norm is replaced by the norm generated by a particular convex body $B$ with a smooth boundary and everywhere non-vanishing curvature. Our construction, based on a combinatorial construction due to Pavel Valtr naturally leads us to some interesting connections between the problem under consideration, geometric incidence theorem in the discrete setting and distribution of lattice points in convex…
We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces on two dimensional vector spaces over finite fields. For higher dimensions, we estimate the decay of the Fourier transform of the characteristic functions on quadratic surfaces so that we obtain the Tomas-Stein exponent. Using incidence theorems, we also study the extension theorems in the restricted settings to sizes of sets in quadratic surfaces. Estimate…
Natural geometric Fourier transforms and the associated fractional Laplacian
本研究针对$\u211d^{n}$上的任意一般几何结构，提出一对相容的几何傅里叶变换，分析其性质与经典傅里叶变换的相容性，并证明正定结构下该变换可诱导性质与经典分数阶拉普拉斯算子相似的几何分数阶拉普拉斯算子。
Incidence estimates for quasi-product sets and applications
本文基于 Furstenberg 集理论，推导了 δ-离散结构的 incidence 结果，并应用于分形测度的能量估计和傅里叶衰减等问题。
In this paper, we prove that the partial sum process of general orthogonal series is a geometric 2-rough process under the same condition as in Menshov-Rademacher Theorem. For Fourier series, the condition can be improved, and an equivalent condition on the limit function is identified.
We prove that if $P,\mathcal{L}$ are finite sets of $δ$-separated points and lines in $\mathbb{R}^{2}$, the number of $δ$-incidences between $P$ and $\mathcal{L}$ is no larger than a constant times $$|P|^{2/3}|\mathcal{L}|^{2/3} \cdot δ^{-1/3}.$$ We apply the bound to obtain the following variant of the Loomis-Whitney inequality in the Heisenberg group: $$ |K| \lesssim |π_{x}(K)|^{2/3} \cdot |π_{y}(K)|^{2/3}, \qquad K \subset \mathbb{H}. $$ Here $π_{x}$ and $π_{y}$ are the vertical projections to the $xt$- and $yt$-planes, respectively, and $|\cdot|$ refers to natural Haar measure on either $\…
Szemerédi-Trotter bounds for tubes and applications
该研究证明了满足填充条件的平面管关联数的Szemerédi-Trotter型精确估计，并将其应用于改进平面曲线上分形测度的傅里叶变换估计。
The aim of this paper is to prove a uniform Fourier restriction estimate for certain $2-$dimensional surfaces in $\mathbb R^{2n}$. These surfaces are the image of complex polynomial curves $γ(z) = (p_1(z), \dots, p_n(z))$, equipped with the complex equivalent to the affine arclength measure. This result is a complex-polynomial counterpart to a previous result by Stovall [Sto16] in the real setting. As a means to prove this theorem we provide an alternative proof of a geometric inequality by Dendrinos and Wright [DW10] that extends the result to complex polynomials.
We consider certain estimates involving averaging operators over curves and hypersurfaces that can be cast into a combinatorial framework. We show that hypersurfaces with nonzero rotational curvature satisfy the usual restricted weak-type bound, but our proof does not involve the Fourier transform. Secondly, we show that a Strichartz-type estimate for the wave equation in 2+1 dimensions can be obtained in a similar fashion, and we give a simplified proof of Wolff's endpoint theorem for maximal averages over circles. Finally, examples are provided that show what the optimal bound can be for the…
A celebrated unit distance conjecture due to Erd\H os says that that the unit distances cannot arise more than $C_εn^{1+ε}$ times (for any $ε>0$) among $n$ points in the Euclidean plane (see e.g. \cite{SST84} and the references contained therein). In three dimensions, the conjectured bound is $Cn^{\frac{4}{3}}$ (see e.g. \cite{KMSS12} and \cite{Z19}). In dimensions four and higher, this problem, in its general formulation, loses meaning because the Lens example shows that one can construct a set of $n$ points in dimension $4$ and higher where the unit distance arises $\approx n^2$ times (see e…
Incidence estimates for $α$-dimensional tubes and $β$-dimensional balls in $\mathbb{R}^2$
本文结合组合与傅里叶分析方法，证明了平面中满足α维间距的δ-管与满足β维间距的δ-球之间的锐利关联估计，并由此得到Furstenberg集大小及离散化和积问题的新下界。
Fourier dimension of conical and cylindrical hypersurfaces
本文证明了由具有非零Gauss曲率的超曲面生成的锥面和柱面的Fourier维数等于其非零主曲率的数量，从而验证了相关猜想，并指出此类曲面非Salem。
In this article we prove a sharp decay estimate for certain multilinear oscillatory integral operators of a form inspired by the general framework of Christ, Li, Tao, and Thiele [6]. A key purpose of this work is to determine when such estimates are stable under smooth perturbations of both the phase and corresponding projections, which are typically only assumed to be linear. The proof is accomplished by a novel decomposition which mixes features of Gabor or windowed Fourier bases with features of wavelet or Littlewood-Paley decompositions. This decomposition very nearly diagonalizes the prob…
The point-plane incidence theorem states that the number of incidences between $n$ points and $m\geq n$ planes in the projective three-space over a field $F$, is $$O\left(m\sqrt{n}+ m k\right),$$ where $k$ is the maximum number of collinear points, with the extra condition $n< p^2$ if $F$ has characteristic $p>0$. This theorem also underlies a state-of-the-art Szemerédi-Trotter type bound for point-line incidences in $F^2$, due to Stevens and de Zeeuw. This review focuses on some recent, as well as new, applications of these bounds that lead to progress in several open geometric questions in $…
We apply geometric incidence estimates in positive characteristic to prove the optimal $L^2 \to L^3$ Fourier extension estimate for the paraboloid in the four-dimensional vector space over a prime residue field. In three dimensions, when $-1$ is not a square, we prove an $L^2 \to L^{\frac{32}{9} }$ extension estimate, improving the previously known exponent $\frac{68}{19}.$
Let $1 \leq k \leq d$ and consider a subset $E\subset \mathbb{R}^d$. In this paper, we study the problem of how large the Hausdorff dimension of $E$ must be in order for the set of distinct noncongruent $k$-simplices in $E$ (that is, noncongruent point configurations of $k+1$ points from $E$) to have positive Lebesgue measure. This generalizes the $k=1$ case, the well-known Falconer distance problem and a major open problem in geometric measure theory. We establish a dimensional lower threshold of $\frac{d(k+1)}{d+2}$ for Falconer type theorems for $k$-simplices. This threshold is nontrivial i…
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