Robust ladders in the HLM tower construction for arithmetic regularity over ${\bf F}_2^{\,n}$

Abstract

Green's arithmetic regularity lemma over $G={\bf F}_2^{\,n}$ has tower-type lower bounds. We show that the order property in the Hosseini--Lovett--Moshkovitz--Shapira construction is robust under adversarial edits. Write $f=s^{-1}\sum_{i=1}^{s}\mathbf 1_{A_i}$, where $s=\lfloor1/(16ε)\rfloor$, and let $d_s$ be the top block dimension. A ladder of length $r$ in a set $A$ is a pattern $a_p+b_q\in A$ exactly when $p\le q$. For every HLM steering system satisfying the construction's spanning conditions, changing $A_s$ on at most $|G|/16$ points leaves a ladder of length $d_s/8$. The proof views the $2^{d_s}$ row traces as a binary code of length $8d_s$ and relative distance greater than $1/4$, then applies Sauer's lemma. We also choose the top steering map so that a superlevel set of $f$ retains a ladder of length at least $d_s/(8\sqrt{s})\ge{\rm twr}(s-2)$ after every edit on at most $ε|G|$ points. Finally, robust ladders obstruct low-index periodic approximation, and sufficiently costly Green regularity forces positive edit distance from every fixed stable class.