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Let $\mathcal{G}(d,n)$ be the Grassmannian manifold of $n$-dimensional subspaces of $\mathbb{R}^{d}$, and let $\pi_{V} \colon \mathbb{R}^{d} \to V$ be the orthogonal projection. We prove that if $\mu$ is a compactly supported Radon measure on $\mathbb{R}^{d}$ satisfying the $s$-dimensional Frostman condition $\mu(B(x,r)) \leq Cr^{s}$ for all $x \in \mathbb{R}^{d}$ and $r > 0$, then $$\int_{\mathcal{G}(d,n)} \|\pi_{V}\mu\|_{L^{p}(V)}^{p} \, d\gamma_{d,n}(V) < \infty, \qquad 1 \leq p < \frac{2d - n - s}{d - s}.$$ The upper bound for $p$ is sharp, at least, for $d - 1 \leq s \leq d$, and every $0 < n < d$. Our motivation for this question comes from finding improved lower bounds on the Hausdorff dimension of $(s,t)$-Furstenberg sets. For $0 \leq s \leq 1$ and $0 \leq t \leq 2$, a set $K \subset \mathbb{R}^{2}$ is called an $(s,t)$-Furstenberg set if there exists a $t$-dimensional family $\mathcal{L}$ of affine lines in $\mathbb{R}^{2}$ such that $\dim_{\mathrm{H}} (K \cap \ell) \geq s$ for all $\ell \in \mathcal{L}$. As a consequence of our projection theorem in $\mathbb{R}^{2}$, we show that every $(s,t)$-Furstenberg set $K \subset \mathbb{R}^{2}$ with $1 < t \leq 2$ satisfies $$\dim_{\mathrm{H}} K \geq 2s + (1 - s)(t - 1).$$ This improves on previous bounds for pairs $(s,t)$ with $s > \tfrac{1}{2}$ and $t \geq 1 + \epsilon$ for a small absolute constant $\epsilon > 0$. We also prove a higher dimensional analogue of this estimate for codimension-1 Furstenberg sets in $\mathbb{R}^{d}$. As another corollary of our method, we obtain a $\delta$-discretised sum-product estimate for $(\delta,s)$-sets. Our bound improves on a previous estimate of Chen for every $\tfrac{1}{2} < s < 1$, and also of Guth-Katz-Zahl for $s \geq 0.5151$., Comment: 28 pages, 3 figures. v3: reviewer comments incorporated, to appear in Adv. Math

We give a lower bound for the size of a subset of $\mathbb F_q^n$ containing a rich k-plane in every direction, a k-plane Furstenberg set. The chief novelty of our method is that we use arguments on non-reduced subschemes and flat families to derive combinatorial facts about incidences between points and k-planes in space., Comment: 16 pages

In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $\alpha,\beta\in(0,1]$, we will say that a set $E\subset \R^2$ is an $F_{\alpha\beta}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $\beta$ and, for each direction $e$ in $L$, there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is equal or greater than $\alpha$. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that $\dim(E)\ge\max\left\{\alpha+\frac{\beta}{2} ; 2\alpha+\beta -1\right\}$ for any $E\in F_{\alpha\beta}$. In particular we are able to extend previously known results to the ``endpoint'' $\alpha=0$ case., Comment: 13 pages

En esta tesis se estudian dos problemas del Análisis Armónico clásico desde el punto de vista de las medidas de Hausdorff. El primero es el problema de Furstenberg, que en su versión clásica se refiere a la determinación de la dimensión de Hausdorff (dim_H ) de los conjuntos de la clase F_α : dado α ∈ (0, 1], un conjunto E ⊆ R^2 está en la clase Fα si para cada e ∈ S existe un segmento unitario l_e en la dirección de e tal que dim_H (l ∩ E) ≥ α. En el caso α = 1, este problema resulta equivalente al problema de Kakeya. Si notamos γ(α) = inf {dim_H (E) : E ∈ Fα }, entonces vale que max {1/2 + α; 2α} ≤ γ(α) ≤ (1 + 3α)/2. (1) En este trabajo se estudia este problema desde una perspectiva más general, en términos de las medidas de Hausdorff h-dimensionales H^h asociadas a funciones de dimensión. Definimos los conjuntos de la clase de Furstenberg F_h asociados a una función h. La hipótesis natural para cada dirección es que H^h (l_e ∩ E) > 0. Generalizamos los resultados conocidos en términos de “saltos logarítmicos” y obtenemos resultados análogos a las cotas clásicas que permiten, además, extender la desigualdad (1) al caso extremo α = 0. Precisamente, se prueba que la función de dimensión apropiada para los conjuntos de la clase F_h no puede ser mucho más chica que h^2 o que la raiz cuadrada de h. Para las cotas superiores exhibimos explícitamente conjuntos en la clase F_h suficientemente chicos. Usamos para eso algunos resultados sobre Aproximación Diofántica,acerca de la dimensión de conjuntos de números “bien aproximables”. Obtenemos resultados acerca de la dimensión de conjuntos en la clase Fαβ, definida como Fα pero sólo para un conjunto L ⊂ S tal que dim_H (L) ≥ β. Probamos una versión de (1) que refleja la interacción entre los parámetros α y β. Este problema fue estudiado también en el conexto general. En segundo lugar se estudió con el mismo enfoque el problema de la Restricción de la Trasformada de Fourier, que se refiere a la posibilidad de darle sentido a la restricción de f a un subconjunto E de R^n . La respuesta depende de la existencia de una medida μ en E con ciertas propiedades de dimensiona- lidad y de decaimiento para su transformada μ. En este contexto se reformuló el teorema de restricción de Stein-Tomas en términos de medidas de Hausdorff. In this thesis we study two problems in classical Harmonic Analysis. The first is the Furstenberg problem, which in its classical form concerns the de termination of the Hausdorff dimension (dim_H) of the sets in the F_α -class: for a given α ∈ (0, 1], a set E ⊆ R^2 is in the Fα-class if for each e ∈ S there exists a unit line segment l_ e in the direction of e such that dim_H (l ∩ E) ≥ α. For α = 1, this problem is essentially equivalent to the “Kakeya needle problem”. If we define γ(α) = inf {dim_H (E) : E ∈ F_α }, then max {1/2 + α; 2α} ≤ γ(α) ≤ (1 + 3α)/2. (1) In this work we approach this problem from a more general point of view, in terms of h-dimensional Hausdorff measures H^h associated to dimension functions. We define the class F_h of Furstenberg sets associated to a given dimension function h. The natural requirement for a set E to belong to F_h , is that H^h (l_e ∩ E) > 0 for each direction. We generalize the known results in terms of ‘logarithmic gaps” and obtain analogues to the estimates given in (1). Moreover, these analogues allow us to extend our results to the endpoint α = 0. Precisely, we prove that the correct dimension function for the class F_h can not be much smaller than h^2 or square root of h. For the upper bounds we exhibit an explicit construction of F_h-sets which are small enough. To that end we prove some results from Diophantine Approximation about the the dimension of the set of “well approximable numbers”. We obtain results about the dimension of Furstenberg sets in the class F_αβ , defined analogously to the class Fα but only for a fractal set L ⊂ S such that dim_H (L) ≥ β. We prove an inequality like (1) which reflects the interplay between α and β. This problem is also studied in the general scenario of Hausdorff measures. The second problem studied in this work, in the same general scenario as before, is the Restriction Problem for the Fourier transform. Here the problem is to give a meaningful sense to the restriction of f to a subset E of R^n . The answer depends on the existence of a measure μ supported on E with precise conditions on the dimensionality of μ and decay properties of μ. In this context we reformulate the Stein-Tomas restriction theorem for Hausdorff measures. Fil: Rela, Ezequiel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

For $\alpha$ in $(0,1]$, a subset $E$ of $\RR$ is called Furstenberg set of type $\alpha$ or $F_\alpha$-set if for each direction $e$ in the unit circle there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is greater or equal than $\alpha$. In this paper we show that if $\alpha > 0$, there exists a set $E\in F_\alpha$ such that $\HH{g}(E)=0$ for $g(x)=x^{1/2+3/2\alpha}\log^{-\theta}(\frac{1}{x})$, $\theta>\frac{1+3\alpha}{2}$, which improves on the the previously known bound, that $H^{\beta}(E) = 0$ for $\beta>1/2+3/2\alpha$. Further, by refining the argument in a subtle way, we are able to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for $\h_\gamma(x)=\log^{-\gamma}(\frac{1}{x})$, $\gamma>0$, we construct a set $E_\gamma\in F_{\h_\gamma}$ of Hausdorff dimension not greater than 1/2. Since in a previous work we showed that 1/2 is a lower bound for the Hausdorff dimension of any $E\in F_{\h_\gamma}$, with the present construction, the value 1/2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions $\h_\gamma$., Comment: Final version

In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional growth conditions on the dimension function, we obtain a lower bound on the dimension of "zero dimensional" Furstenberg sets., 16 pages, 3 figures