F-sets of arbitrary finite width

@misc{giannoni2026fsets,
  title = {{F-sets of arbitrary finite width}},
  author = {Alessandro Giannoni},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05305},
  eprint = {2609.05305},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05305},
}

TY  - UNPB
TI  - F-sets of arbitrary finite width
AU  - Alessandro Giannoni
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.05305
UR  - https://arxiv.org/abs/2609.05305
AB  - Ferraguti and Micheli conjectured that non-trivial F-sets of every prescribed width exist over each finite field. We prove the finite-width part of their conjecture over 𝔽_q for every q≠2,3. Fix a suitable prime ℓ. For a degree bound D, we consider the saturated family of all irreducible polynomials g(X^ℓ^j) whose core g has degree at most D. A factor-descent lemma shows that every irreducible factor arising from a shifted difference belongs to the same family and has strictly smaller core degree. The saturated family is therefore an F-set of finite width. Dirichlet's theorem for 𝔽_q[X] produces core ladders of arbitrary length, and Kummer lifting reproduces each ladder at infinitely many scales. These parallel ladders force the width to be sufficiently large; an appropriate tail of the nullity filtration then has the prescribed width.
ER  -