任意有限宽度的F-集
F-sets of arbitrary finite width
浏览论文内容
中文总结 AI 辅助
本文解决了Ferraguti和Micheli关于有限域上存在任意宽度非平凡F-集的猜想，对q≠2,3的所有情况，构造出了恰好宽度为任意正整数r的无限非平凡F-集，证明了该猜想的有限宽度部分成立。
中文摘要 AI 辅助
Ferraguti和Micheli引入了F-集的宽度，并猜想在每个有限域上都存在任意宽度的非平凡F-集。对于q≠2,3，他们的构造给出了宽度1和2的例子。我们证明，对于每个q≠2,3和每个整数r≥1，在F_q[X]中存在一个无限的非平凡F-集，其宽度恰好为r。因此，在所有这类域上，他们猜想的有限宽度部分已被解决。该证明结合了不可约幂次替换的有界核族、g(Xⁿ)的因式分解结果、F_q[X]上的Dirichlet定理以及Kummer提升。核度给出了宽度的一致上界，而平行后继阶梯给出了所需的下界；零化度滤过的合适尾部具有规定的宽度。
英文摘要
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 $\mathbb{F}_q$ for every $q\neq2,3$. Fix a suitable prime $\ell$. For a degree bound $D$, we consider the saturated family of all irreducible polynomials $g(X^{\ell^j})$ 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 $\mathbb{F}_q[X]$ 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.
