Insensitive.tex

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

\begin{definition} Let $i, j \in [k]$ be distinct, and $I \subseteq [n]$. We say that $A \subseteq [k]^n$ is \emph{$ij$-insensitive on~$I$} if the following condition holds: $x \in A$ iff $\chg{x}{i}{j} \in A$. \end{definition} \begin{remark} This definition is symmetric in $i$ and $j$. It is perhaps easier to understand the condition as follows: ``altering some $i$'s to $j$'s and some $j$'s to $i$'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}