论文标题
恢复功能的仿射线性从其限制到仿射线
Recovering affine-linearity of functions from their restrictions to affine lines
论文作者
论文摘要
Tao-Ziegler的最新结果[离散肛门。 2016年]和Greenfeld-Tao（2022预印本），沿着Abelian组的亚组串联仿生线性函数，我们在恢复功能的仿射线性$ f：v \ t $ w $中从其限制到仿射线的限制，在$ v $ v $ v $ v $ f $ \ natebb f} $ vector $ vector $ nece $ nece $ nece $ vector space和$ diles $中，并显示了三个结果。首先，如果$ \ dim v <| \ mathbb {f} | $和$ f：v \ to \ mathbb {f} $仅限于平行于基础的仿射线，而通过$ 0 $的某些“通用”线路时，则是$ f $ affine-f $ affine linear on $ v $。 （这扩展到所有模块$ m $，超过了Unitital的交换环$ r $具有足够大的特征。这也足以恢复$ V $上的仿射线性，但要达到现场自动形态。特别是，如果$ \ mathbb {f} $是prime字段$ \ mathbb {z}/p \ mathbb {z} $（$ p> 2 $）或$ \ mathbb {q} $，或完成$ \ mathbb {q} _p $或$ \ m mathbb { 然后，我们通过上述第一个结果定量地完善了我们的第一个结果，这是通过Singer [Trans [Trans]最初探索的添加$ B_H $ -SET的弱乘法变体。阿米尔。数学。 Soc。 1938年]，Erdos-turan [J.伦敦数学。 Soc。 1941年和Bose-Chowla [评论。数学。 helv。 1962]。弱乘以$ b_h $ - 集发生在所有环内，具有足够大的特征，并且在所有无限或大的有限积分域/字段中。我们表明，如果$ r $是这些类别的任何类别中的任何类别，而对于某些$ n \ geqslant 3 $，则需要$ m = r^n $，那么人们需要至少$ \ binom {n} {\ lceil n/2 \ rceil} $ affine-linearity，以推断$ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $而且，这种界限很锋利。
Motivated by recent results of Tao-Ziegler [Discrete Anal. 2016] and Greenfeld-Tao (2022 preprint) on concatenating affine-linear functions along subgroups of an abelian group, we show three results on recovering affine-linearity of functions $f : V \to W$ from their restrictions to affine lines, where $V,W$ are $\mathbb{F}$-vector spaces and $\dim V \geqslant 2$. First, if $\dim V < |\mathbb{F}|$ and $f : V \to \mathbb{F}$ is affine-linear when restricted to affine lines parallel to a basis and to certain "generic" lines through $0$, then $f$ is affine-linear on $V$. (This extends to all modules $M$ over unital commutative rings $R$ with large enough characteristic.) Second, we explain how a classical result attributed to von Staudt (1850s) extends beyond bijections: if $f : V \to W$ preserves affine lines $\ell$, and if $f(v) \not\in f(\ell)$ whenever $v \not\in \ell$, then this also suffices to recover affine-linearity on $V$, but up to a field automorphism. In particular, if $\mathbb{F}$ is a prime field $\mathbb{Z}/p\mathbb{Z}$ ($p>2$) or $\mathbb{Q}$, or a completion $\mathbb{Q}_p$ or $\mathbb{R}$, then $f$ is affine-linear on $V$. We then quantitatively refine our first result above, via a weak multiplicative variant of the additive $B_h$-sets initially explored by Singer [Trans. Amer. Math. Soc. 1938], Erdos-Turan [J. London Math. Soc. 1941], and Bose-Chowla [Comment. Math. Helv. 1962]. Weak multiplicative $B_h$-sets occur inside all rings with large enough characteristic, and in all infinite or large enough finite integral domains/fields. We show that if $R$ is among any of these classes of rings, and $M = R^n$ for some $n \geqslant 3$, then one requires affine-linearity on at least $\binom{n}{\lceil n/2 \rceil}$-many generic lines to deduce the global affine-linearity of $f$ on $R^n$. Moreover, this bound is sharp.
