Insensitive.tex

\section{\texorpdfstring{$ab$}{ab}-insensitive sets}

\begin{definition} Let $a, b \in [k]$ be distinct, and $I \subseteq [n]$. We say that $A \subseteq [k]^n$ is \emph{$ab$-insensitive on~$I$} if the following condition holds: $x \in A$ iff $\chg{x}{a}{b} \in A$. \end{definition} \begin{remark} This definition is symmetric in $a$ and $b$. It is perhaps easier to understand the condition as follows: ``altering some $a$'s to $b$'s and some $b$'s to $a$'s does not affect presence/absence in $A$. \end{remark}

\noteryan{Just putting the following statement of subspace-DHJ(k) under product distributions here for the future} \begin{theorem} \label{thm:subsp} Let $d \in \N$, $0 < \eta < 1$, and let $\pi$ be a distribution on $[k]$. Then assuming \[ n \geq n_{\ref{thm:subsp}}(k,d,\eta,\pi) := to be determined, \] every set $A \subseteq [k]^n$ with $\pi^{\otimes n}(A) \geq \eta$ contains a nondegenerate $d$-dimensional subspace. \end{theorem}