English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We prove the following variant of Marstrand's theorem about projections ofcartesian products of sets:  Let $K_1,...,K_n$ Borel subsets of $\mathbb R^{m_1},... ,\mathbb R^{m_n}$respectively, and $\pi:\mathbb R^{m_1}\times...\times\mathbb R^{m_n}\to\mathbbR^k$ be a surjective linear map. We set $$\mathfrak{m}:=\min\{\sum_{i\inI}\dim_H(K_i) + \dim\pi(\bigoplus_{i\in I^c}\mathbb R^{m_i}),I\subset\{1,...,n\}, I\ne\emptyset\}.$$ Consider the space $\Lambda_m=\{(t,O),t\in\mathbb R, O\in SO(m)\}$ with the natural measure and set$\Lambda=\Lambda_{m_1}\times...\times\Lambda_{m_n}$. For every$\lambda=(t_1,O_1,...,t_n,O_n)\in\Lambda$ and every $x=(x^1,,x^n)\in\mathbbR^{m_1}\times...\times\mathbb R^{m_n}$ we define$\pi_\lambda(x)=\pi(t_1O_1x^1,...,t_nO_nx^n)$. Then we have $(i)$ If$\mathfrak{m}>k$, then $\pi_\lambda(K_1\times...\times K_n)$ has positive$k$-dimensional Lebesgue measure for almost every $\lambda\in\Lambda$. $(ii)$If $\mathfrak{m}\leq k$ and $\dim_H(K_1\times...\timesK_n)=\dim_H(K_1)+...+\dim_H(K_n)$, then \dim_H(\pi_\lambda(K_1\times...\timesK_n))=\mathfrak{m}$ for almost every $\lambda\in\Lambda$.

A generalization of Marstrand's theorem for projections of cartesian products