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 Plus annual

Presents the access of premium content as premium feature

Premium Content

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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $G$ be a connected compact group equipped with the normalised Haarmeasure $\mu$. Our first result shows that given $\alpha, \beta>0$, there is aconstant $c = c(\alpha,\beta)>0$ such that for any compact sets $A,B\subseteqG$ with $ \alpha\mu(B)\geq\mu(A)\geq \mu(B) $ and $ \mu(A)+\mu(B)\leq 1-\beta$,there exist $b_1,\dots b_c\in B$ such that \[ \mu(A\cdot \{b_1,\dots,b_c\})\geq\mu(A)+\mu(B).\] A special case of this, that is, when $G=\mathbb{T}^d$,confirms a recent conjecture of Bollob\'as, Leader and Tiba.  We also prove a quantitatively stronger version of such a result in thediscrete setting of $\mathbb{R}^d$. Thus, given $d \in \mathbb{N}$, we showthat there exists $c = c(d) >0$ such that for any finite, non-empty set $A\subseteq \mathbb{R}^d$ which is not contained in a translate of a hyperplane,one can find $a_1, \dots, a_c \in A$ satisfying \[ |A+ \{a_1, \dots, a_c\}|\geq (d+1)|A| - O_d(1). \] The main term here is optimal and recovers thebounds given by Freiman's lemma up to the $O_d(1)$ error term.

Kemperman's inequality and Freiman's lemma via few translates