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Translation-invariant equations in š¯”½pn and groups

  • Katalin Gyarmati
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Abstract

In this paper, we explore how large a subset of a finite group can be without containing non-trivial solutions to linear equations of the form aX+bY=cZ+dU (where a+b=c+d). These equations naturally generalize the classic concept of Sidon sets. Using Tao's slice rank method, we first establish strict upper bounds for the size of such solution-free sets in the vector space š¯”½pn. Next, we turn to cyclic groups ā„¤m and construct surprisingly large sets that have no solutions for the asymmetric equation X+5Y=3U+3Z. These constructions achieve a logarithmic density of roughly $0.5283$, breaking the expected $0.5$ barrier for any sufficiently large modulus m. Finally, we show that this high-density behavior extends to a wider family of equations whose coefficients follow simple rules modulo $8$.

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Cite

BibTeXRIS
BibTeX
@misc{gyarmati2026translationinvariant,
  title = {{Translation-invariant equations in $\mathbbF_p^n$ and groups}},
  author = {Katalin Gyarmati},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09887},
  eprint = {2610.09887},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09887},
}
RIS
TY  - UNPB
TI  - Translation-invariant equations in š¯”½_p^n and groups
AU  - Katalin Gyarmati
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09887
UR  - https://arxiv.org/abs/2610.09887
AB  - In this paper, we explore how large a subset of a finite group can be without containing non-trivial solutions to linear equations of the form aX+bY=cZ+dU (where a+b=c+d). These equations naturally generalize the classic concept of Sidon sets. Using Tao's slice rank method, we first establish strict upper bounds for the size of such solution-free sets in the vector space š¯”½_p^n. Next, we turn to cyclic groups ā„¤_m and construct surprisingly large sets that have no solutions for the asymmetric equation X+5Y=3U+3Z. These constructions achieve a logarithmic density of roughly $0.5283$, breaking the expected $0.5$ barrier for any sufficiently large modulus m. Finally, we show that this high-density behavior extends to a wider family of equations whose coefficients follow simple rules modulo $8$.
ER  -