PreprintCC BY 4.0arXiv
Robust order properties in the HLM construction
Abstract
Green's arithmetic regularity lemma over G=F2n has tower-type lower bounds. We show that the large order properties in the Hosseini--Lovett--Moshkovitz--Shapira (HLM) construction survive sparse edits. If d is the top block dimension, then changing at most |G|/16 points of the top HLM set As still leaves a half-graph of height d/8. For a suitable choice of the HLM maps, changing at most eps|G| points of a superlevel set of the averaged witness still leaves a half-graph of height d/(8√s)≥twr(s−2). The proofs use Sauer's lemma and the binary code formed by the top-level row traces.
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BibTeX
@misc{menezes2026robust,
title = {{Robust order properties in the HLM construction}},
author = {Dean Matthew Menezes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.19015},
eprint = {2609.19015},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19015},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Robust order properties in the HLM construction
AU - Dean Matthew Menezes
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2609.19015
UR - https://arxiv.org/abs/2609.19015
AB - Green's arithmetic regularity lemma over G=F_2^n has tower-type lower bounds. We show that the large order properties in the Hosseini--Lovett--Moshkovitz--Shapira (HLM) construction survive sparse edits. If d is the top block dimension, then changing at most |G|/16 points of the top HLM set A_s still leaves a half-graph of height d/8. For a suitable choice of the HLM maps, changing at most eps|G| points of a superlevel set of the averaged witness still leaves a half-graph of height d/(8√s)≥twr(s−2). The proofs use Sauer's lemma and the binary code formed by the top-level row traces.
ER -