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We shall explain here an idea to generalize classical complex analytic Kleinian group theory to any odd-dimensional cases. For a certain class of discrete subgroups of 
$\text{PGL}_{2n+1}(\mathbf{C})$ acting on 
$\mathbf{P}^{2n+1}$, we can define their domains of discontinuity in a canonical manner, regarding an 
$n$-dimensional projective linear subspace in 
$\mathbf{P}^{2n+1}$ as a point, like a point in the classical one-dimensional case. Many interesting (compact) non-Kähler manifolds appear systematically as the canonical quotients of the domains. In the last section, we shall give some examples.
For 
$x\in (0,1]$ and a positive integer 
$n,$ let 
$S_{\!n}(x)$ denote the summation of the first 
$n$ digits in the dyadic expansion of 
$x$ and let 
$r_{n}(x)$ denote the run-length function. In this paper, we obtain the Hausdorff dimensions of the following sets: where 
$$\begin{eqnarray}\bigg\{x\in (0,1]:\liminf _{n\rightarrow \infty }\frac{S_{\!n}(x)}{n}=\unicode[STIX]{x1D6FC},\limsup _{n\rightarrow \infty }\frac{S_{\!n}(x)}{n}=\unicode[STIX]{x1D6FD},\lim _{n\rightarrow \infty }\frac{r_{n}(x)}{\log _{2}n}=\unicode[STIX]{x1D6FE}\bigg\},\end{eqnarray}$$
$0\leq \unicode[STIX]{x1D6FC}\leq \unicode[STIX]{x1D6FD}\leq 1$, 
$0\leq \unicode[STIX]{x1D6FE}\leq +\infty$.
Let 
$I(n)$ denote the number of isomorphism classes of subgroups of 
$(\mathbb{Z}/n\mathbb{Z})^{\times }$, and let 
$G(n)$ denote the number of subgroups of 
$(\mathbb{Z}/n\mathbb{Z})^{\times }$ counted as sets (not up to isomorphism). We prove that both 
$\log G(n)$ and 
$\log I(n)$ satisfy Erdős–Kac laws, in that suitable normalizations of them are normally distributed in the limit. Of note is that 
$\log G(n)$ is not an additive function but is closely related to the sum of squares of additive functions. We also establish the orders of magnitude of the maximal orders of 
$\log G(n)$ and 
$\log I(n)$.
We establish existence of weighted Hardy and Rellich inequalities on the spaces 
$L_{p}(\unicode[STIX]{x1D6FA})$, where 
$\unicode[STIX]{x1D6FA}=\mathbf{R}^{d}\backslash K$ with 
$K$ a closed convex subset of 
$\mathbf{R}^{d}$. Let 
$\unicode[STIX]{x1D6E4}=\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ denote the boundary of 
$\unicode[STIX]{x1D6FA}$ and 
$d_{\unicode[STIX]{x1D6E4}}$ the Euclidean distance to 
$\unicode[STIX]{x1D6E4}$. We consider weighting functions 
$c_{\unicode[STIX]{x1D6FA}}=c\circ d_{\unicode[STIX]{x1D6E4}}$ with 
$c(s)=s^{\unicode[STIX]{x1D6FF}}(1+s)^{\unicode[STIX]{x1D6FF}^{\prime }-\unicode[STIX]{x1D6FF}}$ and 
$\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\prime }\geq 0$. Then the Hardy inequalities take the form and the Rellich inequalities are given by 
$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}c_{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}\unicode[STIX]{x1D711}|^{p}\geq b_{p}\int _{\unicode[STIX]{x1D6FA}}c_{\unicode[STIX]{x1D6FA}}\,d_{\unicode[STIX]{x1D6E4}}^{-p}|\unicode[STIX]{x1D711}|^{p}\end{eqnarray}$$ with 
$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}|H\unicode[STIX]{x1D711}|^{p}\geq d_{p}\int _{\unicode[STIX]{x1D6FA}}|c_{\unicode[STIX]{x1D6FA}}\,d_{\unicode[STIX]{x1D6E4}}^{-2}\unicode[STIX]{x1D711}|^{p}\end{eqnarray}$$
$H=-\text{div}(c_{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FB})$. The constants 
$b_{p},d_{p}$ depend on the weighting parameters 
$\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\prime }\geq 0$ and the Hausdorff dimension of the boundary. We compute the optimal constants in a broad range of situations.
Given a vector field on a manifold 
$M$, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved quantities are well behaved under transgression to spaces of maps into 
$M$. We focus on the case of multisymplectic manifolds and Hamiltonian vector fields. Our main result is that in the presence of a Lie group of symmetries admitting a homotopy co-momentum map, one obtains a whole family of globally conserved quantities. This extends a classical result in symplectic geometry. We carry this out in a general setting, considering several variants of the notion of globally conserved quantity.