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Simultaneous popular differences for polynomial and linear progressions in finite fields

  • Xuancheng Shao
  • Weiwen Zhang
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Abstract

We study simultaneous popular differences for polynomial progressions and linear progressions over finite fields 𝔽p. For certain families of polynomial progressions of algebraic complexity $1$, such as (x, x+y, x+2y) and (x, x+y3, x+2y3), we prove the existence of y ≠ 0 which is popular simultaneously for both progressions. This extends previous results by Conlon, Dong and Hong who considered complexity $0$ progressions. On the other hand, we construct counterexamples showing that simultaneous popular differences are not guaranteed to exist for (x, x+y, x+2y) and (x, x+y2, x+2y2), or for (x, x+y, x+2y) and (x, x+y, x+3y), or for (x, x+y, x+2y) and (x,x+y,x+2y,x+3y), answering questions posed by Conlon, Dong and Hong.

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Cite

BibTeXRIS
BibTeX
@misc{shao2026simultaneous,
  title = {{Simultaneous popular differences for polynomial and linear progressions in finite fields}},
  author = {Xuancheng Shao and Weiwen Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28226},
  eprint = {2610.28226},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28226},
}
RIS
TY  - UNPB
TI  - Simultaneous popular differences for polynomial and linear progressions in finite fields
AU  - Xuancheng Shao
AU  - Weiwen Zhang
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28226
UR  - https://arxiv.org/abs/2610.28226
AB  - We study simultaneous popular differences for polynomial progressions and linear progressions over finite fields 𝔽_p. For certain families of polynomial progressions of algebraic complexity $1$, such as (x, x+y, x+2y) and (x, x+y^3, x+2y^3), we prove the existence of y ≠ 0 which is popular simultaneously for both progressions. This extends previous results by Conlon, Dong and Hong who considered complexity $0$ progressions. On the other hand, we construct counterexamples showing that simultaneous popular differences are not guaranteed to exist for (x, x+y, x+2y) and (x, x+y^2, x+2y^2), or for (x, x+y, x+2y) and (x, x+y, x+3y), or for (x, x+y, x+2y) and (x,x+y,x+2y,x+3y), answering questions posed by Conlon, Dong and Hong.
ER  -