Let gamma: [0,1] -> S-2 be a non-degenerate curve in R-3, that is to say, det(gamma(0), gamma '(0), gamma ''(0)) =/ 0. For each 0 is an element of [0, 1], let V theta = gamma (0)perpendicular to and let pi(theta) : R-3 -> V-theta be the orthogonal projections. We prove that if A subset of R-3 is a Borel set, then for a.e. 0 is an element of [0,1] we have dim(pi(theta)(A)) = min{2, dimA}. More generally, we prove an exceptional set estimate. For A subset of R-3 and 0 <= s <= 2, define Es(A) := {0 is an element of [0,1] : dim(pi(theta) (A)) < s}. We have dim(Es(A)) <= max{1 + s-dim(A), 0}. We also prove that if dim(A) > 2, then for a.e. 0 is an element of [0,1] we have H-2(pi(theta) (A)) > 0.
abstract: Let $\gamma:[0,1]\rightarrow\mathbb{S}^2$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det(\gamma(\theta),\gamma'(\theta),\linebreak\gamma" (\theta))\neq 0$. For each $\theta\in[0,1]$, let $V_\theta=\gamma(\theta)^\perp$ and let $\pi_\theta:\mathbb{R}^3\rightarrow V_\theta$ be the orthogonal projections. We prove that if $A\subset\mathbb{R}^3$ is a Borel set, then for a.e. $\theta\in [0,1]$ we have $\dim(\pi_\theta(A))=\min\{2,\dim A\}$. More generally, we prove an exceptional set estimate. For $A\subset\mathbb{R}^3$ and $0\le s\le 2$, define $E_s(A):=\{\theta\in[0,1]:\dim(\pi_\theta(A))2$, then for a.e. $\theta\in[0,1]$ we have $\mathcal{H}^2(\pi_\theta(A))>0$.
abstract: The proofs of Theorem 1.1 and Theorem 1.5\,(2) in the authors' paper \emph{The Hasse norm principle for abelian extensions} are incorrect. We point out the mistakes and provide correct proofs, using techniques of the original paper.
The theta correspondence for the dual pair (Uen,Uep,q) (where covers of Un and Up,q respectively) is used to find nice bases for a collection of unitary highest weight modules of Uep,q, and also for the highest weight vectors with respect to the obvious maximal compact subgroup. The techniques used involve signed Hibi rings, an extension of the concept of Hibi ring, which has been identified as an underlying structure in constructing standard monomial theories, and in finding straightening laws in certain rings. An extension of the Littlewood-Richardson rule to tensor products of arbitrary rational representations of GLn(C) is required for finding the desired bases, and is also provided. Uen and Uep,q are 2-fold
Let m >= 3, we prove that (alpha n(theta) mod 1)n>0 has Poissonian m-point correlation for all alpha > 0, provided theta
