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: for all $x \in A$, if $x'$ is formed by changing $x_i$ from $a$ to $b$ or $b$ to $a$ for some $i \in I$, then $x' \in A$ also. If $I = [n]$ we simply say that $A$ is \emph{$ab$-insensitive}.\noteryan{Perhaps use the $\chg{x}{a}{b}$ notation; e.g., $ab$-insens.\ means $x \in A$ iff $\chg{x}{a}{b} \in A$} \end{definition}

\noteryan{Just putting this 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}