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

We present a unified framework to study threshold functions for the existenceof solutions to linear systems of equations in random sets which includesarithmetic progressions, sum-free sets, $B_{h}[g]$-sets and Hilbert cubes. Inparticular, we show that there exists a threshold function for the property"$\mathcal{A}$ contains a non-trivial solution of$M\cdot\textbf{x}=\textbf{0}$", where $\mathcal{A}$ is a random set and each ofits elements is chosen independently with the same probability from theinterval of integers $\{1,\dots,n\}$. Our study contains a formal definition oftrivial solutions for any combinatorial structure, extending a previousdefinition by Ruzsa when dealing with a single equation.  Furthermore, we study the behaviour of the distribution of the number ofnon-trivial solutions at the threshold scale. We show that it converges to aPoisson distribution whose parameter depends on the volumes of certain convexpolytopes arising from the linear system under study as well as the symmetryinherent in the structures, which we formally define and characterize.

Threshold functions and Poisson convergence for systems of equations in random sets