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

In this paper, we prove the following two results. Let $d$ be a natural number and $q,s$ be co-prime integers such that $1 0$ depending only on $q,s$ and $d$ such that for any finite subset $A$ of $\mathbb{R}^d$ that is not contained in a translate of a hyperplane, we have $$ |q\cdot A + s\cdot A| \geq (|q| +|s|+ 2d-2)|A| - O_{q,s,d}(|A|^{1-\delta}) . $$ The main term in this bound is sharp and improves upon an earlier result of Balog and Shakan. Secondly, let $\mathscr{L} \in \textrm{GL}_{2}( \mathbb{R})$ be a linear transformation such that $\mathscr{L}$ does not have any invariant one-dimensional subspace of $\mathbb{R}^2$. Then for all finite subsets $A$ of $\mathbb{R}^2$, we have $$ |A + \mathscr{L}(A)| \geq 4|A| - O(|A|^{1-\delta}), $$ for some absolute constant $\delta > 0$. The main term in this result is sharp as well.

Sums of Linear Transformations in Higher Dimensions