Abstract:
In this paper, we prove some generalizations of results concerning the Eneström-Kakeya theorem. The results obtained considerably improve the bounds by relaxing the hypothesis in some cases.

Abstract:
In this paper we present some interesting generalizations of Enestr？m-Kakeya type results concerning the location of zeros of a polynomial in the complex plane. We relax the hypothesis and put less restrictive conditions on the coefficients of the polynomial, and thereby generalize some classical results. DOI: http://dx.doi.org/10.3126/bibechana.v10i0.8067 ？ BIBECHANA 10 (2014) 71-81

Abstract:
We give an essentially self-contained proof of Guth's recent endpoint multilinear Kakeya theorem which avoids the use of somewhat sophisticated algebraic topology, and which instead appeals to the Borsuk-Ulam theorem.

Abstract:
This research paper concentrates on
the Kakeya problem. After the introduction of historical issue, we provide a
thorough presentation of the results of Kakeya problem with some examples of
the early solutions as well as the proof of the final outcome of this problem,
the solution of which is known as Besicovitch Set. We give 3 different
construction of Besicovitch set as well as the intuition of construction, which
is related to iterated integral of 2-variable real function. We also give the
Cunningham construction in which the area of a simply connected Kakeya set can
also tend to 0. Furthermore, we generalize the process of generating a Kakeya
set into a Kakeya dynamic. The definition of multiplicity enables us to
estimate the area of a Kakeya set. In following discussion we provided a
conjecture related to the solution in particular range. Finally, the derivation
of the Kakeya problem is presented.

Abstract:
This thesis investigates two problems that are discrete analogues of two harmonic analytic problems which lie in the heart of research in the field. More specifically, we consider discrete analogues of the maximal Kakeya operator conjecture and of the recently solved endpoint multilinear Kakeya problem, by effectively shrinking the tubes involved in these problems to lines, thus giving rise to the problems of counting joints and multijoints with multiplicities. In fact, we effectively show that, in $\mathbb{R}^3$, what we expect to hold due to the maximal Kakeya operator conjecture, as well as what we know in the continuous case due to the endpoint multilinear Kakeya theorem by Guth, still hold in the discrete case. In particular, let $\mathfrak{L}$ be a collection of $L$ lines in $\mathbb{R}^3$ and $J$ the set of joints formed by $\mathfrak{L}$, that is, the set of points each of which lies in at least three non-coplanar lines of $\mathfrak{L}$. It is known that $|J|=O(L^{3/2})$ (first proved by Guth and Katz). For each joint $x\in J$, let the multiplicity $N(x)$ of $x$ be the number of triples of non-coplanar lines through $x$. We prove here that $$\sum_{x\in J} N(x)^{1/2}=O(L^{3/2}), $$while we also extend this result to real algebraic curves in $\mathbb{R}^3$ of uniformly bounded degree, as well as to curves in $\mathbb{R}^3$ parametrized by real univariate polynomials of uniformly bounded degree. The multijoints problem is a variant of the joints problem, involving three finite collections of lines in $\mathbb{R}^3$; a multijoint formed by them is a point that lies in (at least) three non-coplanar lines, one from each collection. We finally present some results regarding the joints problem in different field settings and higher dimensions.

Abstract:
The purpose of this article is to survey the developments on the Kakeya problem in recent years, concentrating on the period after the excellent 1999 survey of Wolff, and including some recent work by the authors. We will focus on the standard Kakeya problem for line segments and not discuss other important variants (such as Kakeya estimates for circles, light rays, or $k$-planes).

Abstract:
We completely characterize the boundedness of planar directional maximal operators on L^p. More precisely, if Omega is a set of directions, we show that M_Omega, the maximal operator associated to line segments in the directions Omega, is unbounded on L^p, for all p < infinity, precisely when Omega admits Kakeya-type sets. In fact, we show that if Omega does not admit Kakeya sets, then Omega is a generalized lacunary set, and hence M_Omega is bounded on L^p, for p>1.

Abstract:
We establish new estimates on the Minkowski and Hausdorff dimensions of Besicovitch sets and obtain new bounds on the Kakeya maximal operator.