Arithmetic Progressions in Midpoint Colourings
Abstract
We introduce an asymmetric variant of a classic problem by Roth about colourings of integers and midpoints. The new problem cannot be solved with the standard approach by Erdös-Sárközy-Sós which uses symmetry. We describe a Fourier Analytic approach which gives asymptotically tight bounds. In the 𝔽3n setting, the density increment we show is efficient and together with the Freiman-Ruzsa Theorem provides an affine subspace of near optimal codimension within the distinguished set. This approach adopted to the setting of the integers from $1$ to N via Bohr sets and Bogolyubov-Ruzsa Lemma gives a progression of length being a power of N which only depends on the number of colours. We then use Chang's Lemma and Balog-Szemerédi-Gowers Theorem to further improve the dependence on the number of colours.
Related
- Triple divisor type functions in arithmetic progressions to prime power moduli
- Hitting Arithmetic Progressions at the Square-Root Scale
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BibTeX
@misc{kosciuszko2026arithmetic,
title = {{Arithmetic Progressions in Midpoint Colourings}},
author = {Tomasz Kościuszko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11847},
eprint = {2610.11847},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11847},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Arithmetic Progressions in Midpoint Colourings
AU - Tomasz Kościuszko
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11847
UR - https://arxiv.org/abs/2610.11847
AB - We introduce an asymmetric variant of a classic problem by Roth about colourings of integers and midpoints. The new problem cannot be solved with the standard approach by Erdös-Sárközy-Sós which uses symmetry. We describe a Fourier Analytic approach which gives asymptotically tight bounds. In the 𝔽_3^n setting, the density increment we show is efficient and together with the Freiman-Ruzsa Theorem provides an affine subspace of near optimal codimension within the distinguished set. This approach adopted to the setting of the integers from $1$ to N via Bohr sets and Bogolyubov-Ruzsa Lemma gives a progression of length being a power of N which only depends on the number of colours. We then use Chang's Lemma and Balog-Szemerédi-Gowers Theorem to further improve the dependence on the number of colours.
ER -